Difference between revisions of "Sums"

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(→‎Sum of Natural Numbers)
 
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Line 4: Line 4:
  
 
=== even number of terms = last term is even ===
 
=== even number of terms = last term is even ===
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^6 i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: 1-6]]
 +
}}</div></div><br>
  
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^8 i</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^8 i</math><div class="mw-collapsible-content">
Line 15: Line 21:
 
[[category: #sum of natural numbers]]
 
[[category: #sum of natural numbers]]
 
[[comment: 1-10]]
 
[[comment: 1-10]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=4}^{10} i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: 4-10]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=5}^{10} i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: 5-10]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=6}^{11} i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: 6-11]]
 
}}</div></div><br>
 
}}</div></div><br>
  
Line 21: Line 45:
 
[[category: #sum of natural numbers]]
 
[[category: #sum of natural numbers]]
 
[[comment: 1-12]]
 
[[comment: 1-12]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{14} i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: 1-14]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{16} i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: 1-16]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{18} i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: 1-18]]
 
}}</div></div><br>
 
}}</div></div><br>
  
Line 27: Line 69:
 
[[category: #sum of natural numbers]]
 
[[category: #sum of natural numbers]]
 
[[comment: 1-20]]
 
[[comment: 1-20]]
−
}}
+
}}</div></div><br>
  
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{100} i</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{100} i</math><div class="mw-collapsible-content">
Line 34: Line 76:
 
[[comment: 1-100]]
 
[[comment: 1-100]]
 
}}</div></div><br>
 
}}</div></div><br>
−
 
  
 
=== odd number of terms = last term is odd ===
 
=== odd number of terms = last term is odd ===
Line 48: Line 89:
 
[[category: #sum of natural numbers]]
 
[[category: #sum of natural numbers]]
 
[[comment: 1-9]]
 
[[comment: 1-9]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{11} i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: 1-11]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{13} i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: 1-13]]
 
}}</div></div><br>
 
}}</div></div><br>
  
Line 60: Line 113:
 
[[category: #sum of natural numbers]]
 
[[category: #sum of natural numbers]]
 
[[comment: 1-19]]
 
[[comment: 1-19]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{25} i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: 1-25]]
 +
}}</div></div><br>
 +
 +
=== Find the last term ===
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} i=55</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: aₙ, 1, 55]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=5}^{n} i=45</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: aₙ, 5, 45]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} i=120</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: aₙ, 1, 120]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} i=210</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: aₙ, 1, 210]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} i=465</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: aₙ, 1, 465]]
 +
}}</div></div><br>
 +
 +
=== Find the first term ===
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=m}^{10} i=45</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of natural numbers]]
 +
[[comment: a₁, 10, 45]]
 
}}</div></div><br>
 
}}</div></div><br>
  
Line 68: Line 167:
 
[[category: #sum of evens]]
 
[[category: #sum of evens]]
 
[[comment: 2-8]]
 
[[comment: 2-8]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{5} 2i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of evens]]
 +
[[comment: 2-10]]
 
}}</div></div><br>
 
}}</div></div><br>
  
Line 74: Line 179:
 
[[category: #sum of evens]]
 
[[category: #sum of evens]]
 
[[comment: 2-12]]
 
[[comment: 2-12]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=3}^{6} 2i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of evens]]
 +
[[comment: 6-12]]
 
}}</div></div><br>
 
}}</div></div><br>
  
Line 80: Line 191:
 
[[category: #sum of evens]]
 
[[category: #sum of evens]]
 
[[comment: 2-20]]
 
[[comment: 2-20]]
 +
}}</div></div><br>
 +
 +
=== Find the last term ===
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} 2i=42</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of evens]]
 +
[[comment: aₙ, 2, 42]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} 2i=30</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of evens]]
 +
[[comment: aₙ, 2, 30]]
 
}}</div></div><br>
 
}}</div></div><br>
  
Line 102: Line 227:
 
}}</div></div><br>
 
}}</div></div><br>
  
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=4}^{7} \left(2i-1\right)</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of odds]]
 +
[[comment: 7-13]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{15} \left(2i-1\right)</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of odds]]
 +
[[comment: 1-15]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{10} \left(2i-1\right)</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of odds]]
 +
[[comment: 1-19]]
 +
}}</div></div><br>
 +
 +
=== Find the last term ===
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} \left(2i-1\right)=25</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of odds]]
 +
[[comment: aₙ, 1, 25]]
 +
}}</div></div><br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} \left(2i-1\right)=36</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of odds]]
 +
[[comment: aₙ, 1, 36]]
 +
}}</div></div><br>
 +
 +
== Sum of Sevens ==
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^9 7i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum]]
 +
[[comment: 7i,1-9]]
 +
}}</div></div><br>
 +
 +
== Sum of Triples==
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{7} 3i</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum[0] ]]
 +
[[comment: 3i, 1-7]]
 +
}}</div></div><br>
  
 
== Sum of Powers of Two = Sum of Even-Times-Even Numbers ==
 
== Sum of Powers of Two = Sum of Even-Times-Even Numbers ==
Line 121: Line 293:
 
[[category: #sum of powers of two]]
 
[[category: #sum of powers of two]]
 
[[comment: 1-128]]
 
[[comment: 1-128]]
−
}}
+
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{12} 2^{i-1}</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of powers of two]]
 +
[[comment: 1-4095]]
 +
}}</div></div><br>
 +
 
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=3}^{6} 2^{i-1}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=3}^{6} 2^{i-1}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
Line 130: Line 309:
 
== Sum of Powers of Three ==
 
== Sum of Powers of Three ==
  
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{4} 3^{i-1}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
 
[[category: #sum of powers of three]]
 
[[category: #sum of powers of three]]
 
[[comment: 1-27]]
 
[[comment: 1-27]]
−
}}
+
}}</div></div><br>
  
 
== Sum of Powers of Four ==
 
== Sum of Powers of Four ==
  
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{3} 4^{i-1}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
 
[[category: #sum of powers of four]]
 
[[category: #sum of powers of four]]
 
[[comment: 1-16]]
 
[[comment: 1-16]]
−
}}
+
}}</div></div><br>
 +
 
 +
== Sum of Squares ==
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{4} i^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of squares]]
 +
[[comment: 1-4]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{5} i^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of squares]]
 +
[[comment: 1-5]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{7} i^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of squares]]
 +
[[comment: 1-7]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=4}^{8} i^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of squares]]
 +
[[comment: 4-8]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{10} i^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of squares]]
 +
[[comment: 1-10]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{12} i^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of squares]]
 +
[[comment: 1-12]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=5}^{12} i^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of squares]]
 +
[[comment: 5-12]]
 +
}}</div></div><br>
 +
 
 +
=== Squares of Evens ===
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{4} \left(2i\right)^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #squares of evens]]
 +
[[comment: 2-8]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{5} \left(2i\right)^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #squares of evens]]
 +
[[comment: 2-10]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=3}^{8} \left(2i\right)^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #squares of evens]]
 +
[[comment: 6-16]]
 +
}}</div></div><br>
 +
 
 +
=== Squares of Odds ===
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{4} \left(2i-1\right)^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #squares of odds]]
 +
[[comment: 1-7]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{5} \left(2i-1\right)^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #squares of odds]]
 +
[[comment: 1-9]]
 +
}}</div></div><br>
 +
 
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{4} \left(4i\right)^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of squares]]
 +
[[comment: 4i,1-7]]
 +
}}</div></div><br>
 +
 
 +
== Sum of Cubes ==
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{5} i^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of cubes]]
 +
[[comment: 1-5]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{6} i^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of cubes]]
 +
[[comment: 1-6]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{7} i^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of cubes]]
 +
[[comment: 1-7]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{10} i^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of cubes]]
 +
[[comment: 1-10]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=4}^{10} i^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of cubes]]
 +
[[comment: 4-10]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=5}^{10} i^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of cubes]]
 +
[[comment: 5-10]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=5}^{12} i^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of cubes]]
 +
[[comment: 5-12]]
 +
}}</div></div><br>
 +
 
 +
=== Find the last term ===
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} i^3=225</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of cubes]]
 +
[[comment: aₙ, 1, 225]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} i^3=3025</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of cubes]]
 +
[[comment: aₙ, 1, 3025]]
 +
}}</div></div><br>
 +
 
 +
=== Cubes of Odds ===
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{5} \left(2i-1\right)^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #cubes of odds]]
 +
[[comment: 1-9]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{7} \left(2i-1\right)^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #cubes of odds]]
 +
[[comment: 1-13]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=3}^{5} \left(2i-1\right)^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #cubes of odds]]
 +
[[comment: 5-9]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=4}^{6} \left(2i-1\right)^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #cubes of odds]]
 +
[[comment: 7-11]]
 +
}}</div></div><br>
 +
 
 +
==== Find the last term ====
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} \left(2i-1\right)^3=256</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #cubes of odds]]
 +
[[comment: aₙ, 1, 256]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} \left(2i-1\right)^3=1225</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #cubes of odds]]
 +
[[comment: aₙ, 1, 1225]]
 +
}}</div></div><br>
 +
 
 +
=== Cubes of Evens ===
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{3} \left(2i\right)^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #cubes of evens]]
 +
[[comment: 2-6]]
 +
}}</div></div><br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{5} \left(2i\right)^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #cubes of evens]]
 +
[[comment: 2-10]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{6} \left(2i\right)^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #cubes of evens]]
 +
[[comment: 2-12]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=2}^{6} \left(2i\right)^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #cubes of evens]]
 +
[[comment: 4-12]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=4}^{8} \left(2i\right)^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #cubes of evens]]
 +
[[comment: 8-16]]
 +
}}</div></div><br>
 +
 
 +
==== Find the last term ====
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} \left(2i\right)^3=400</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #cubes of evens]]
 +
[[comment: aₙ, 2, 400]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{n} \left(2i\right)^3=1800</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #cubes of evens]]
 +
[[comment: aₙ, 2, 1800]]
 +
}}</div></div><br>
 +
 
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sum_{i=1}^{5} \left(4i\right)^3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #sum of cubes]]
 +
[[comment: 4i,1-5]]
 +
}}</div></div><br>

Latest revision as of 16:24, 1 July 2023

Sum of Natural Numbers

even number of terms = last term is even

[Expand]\scriptstyle\sum_{i=1}^6 i

[Expand]\scriptstyle\sum_{i=1}^8 i

[Expand]\scriptstyle\sum_{i=1}^{10} i

[Expand]\scriptstyle\sum_{i=4}^{10} i

[Expand]\scriptstyle\sum_{i=5}^{10} i

[Expand]\scriptstyle\sum_{i=6}^{11} i

[Expand]\scriptstyle\sum_{i=1}^{12} i

[Expand]\scriptstyle\sum_{i=1}^{14} i

[Expand]\scriptstyle\sum_{i=1}^{16} i

[Expand]\scriptstyle\sum_{i=1}^{18} i

[Expand]\scriptstyle\sum_{i=1}^{20} i

[Expand]\scriptstyle\sum_{i=1}^{100} i

odd number of terms = last term is odd

[Expand]\scriptstyle\sum_{i=1}^{7} i

[Expand]\scriptstyle\sum_{i=1}^{9} i

[Expand]\scriptstyle\sum_{i=1}^{11} i

[Expand]\scriptstyle\sum_{i=1}^{13} i

[Expand]\scriptstyle\sum_{i=1}^{17} i

[Expand]\scriptstyle\sum_{i=1}^{19} i

[Expand]\scriptstyle\sum_{i=1}^{25} i

Find the last term

[Expand]\scriptstyle\sum_{i=1}^{n} i=55

[Expand]\scriptstyle\sum_{i=5}^{n} i=45

[Expand]\scriptstyle\sum_{i=1}^{n} i=120

[Expand]\scriptstyle\sum_{i=1}^{n} i=210

[Expand]\scriptstyle\sum_{i=1}^{n} i=465

Find the first term

[Expand]\scriptstyle\sum_{i=m}^{10} i=45

Sum of Evens

[Expand]\scriptstyle\sum_{i=1}^{4} 2i

[Expand]\scriptstyle\sum_{i=1}^{5} 2i

[Expand]\scriptstyle\sum_{i=1}^{6} 2i

[Expand]\scriptstyle\sum_{i=3}^{6} 2i

[Expand]\scriptstyle\sum_{i=1}^{10} 2i

Find the last term

[Expand]\scriptstyle\sum_{i=1}^{n} 2i=42

[Expand]\scriptstyle\sum_{i=1}^{n} 2i=30

Sum of Odds

[Expand]\scriptstyle\sum_{i=1}^{5} \left(2i-1\right)

[Expand]\scriptstyle\sum_{i=1}^{6} \left(2i-1\right)

[Expand]\scriptstyle\sum_{i=1}^{7} \left(2i-1\right)

[Expand]\scriptstyle\sum_{i=4}^{7} \left(2i-1\right)

[Expand]\scriptstyle\sum_{i=1}^{15} \left(2i-1\right)

[Expand]\scriptstyle\sum_{i=1}^{10} \left(2i-1\right)

Find the last term

[Expand]\scriptstyle\sum_{i=1}^{n} \left(2i-1\right)=25

[Expand]\scriptstyle\sum_{i=1}^{n} \left(2i-1\right)=36

Sum of Sevens

[Expand]\scriptstyle\sum_{i=1}^9 7i

Sum of Triples

[Expand]\scriptstyle\sum_{i=1}^{7} 3i

Sum of Powers of Two = Sum of Even-Times-Even Numbers

[Expand]\scriptstyle\sum_{i=1}^{6} 2^{i-1}

[Expand]\scriptstyle\sum_{i=1}^{7} 2^{i-1}

[Expand]\scriptstyle\sum_{i=1}^{8} 2^{i-1}

[Expand]\scriptstyle\sum_{i=1}^{12} 2^{i-1}

[Expand]\scriptstyle\sum_{i=3}^{6} 2^{i-1}

Sum of Powers of Three

[Expand]\scriptstyle\sum_{i=1}^{4} 3^{i-1}

Sum of Powers of Four

[Expand]\scriptstyle\sum_{i=1}^{3} 4^{i-1}

Sum of Squares

[Expand]\scriptstyle\sum_{i=1}^{4} i^2

[Expand]\scriptstyle\sum_{i=1}^{5} i^2

[Expand]\scriptstyle\sum_{i=1}^{7} i^2

[Expand]\scriptstyle\sum_{i=4}^{8} i^2

[Expand]\scriptstyle\sum_{i=1}^{10} i^2

[Expand]\scriptstyle\sum_{i=1}^{12} i^2

[Expand]\scriptstyle\sum_{i=5}^{12} i^2

Squares of Evens

[Expand]\scriptstyle\sum_{i=1}^{4} \left(2i\right)^2

[Expand]\scriptstyle\sum_{i=1}^{5} \left(2i\right)^2

[Expand]\scriptstyle\sum_{i=3}^{8} \left(2i\right)^2

Squares of Odds

[Expand]\scriptstyle\sum_{i=1}^{4} \left(2i-1\right)^2

[Expand]\scriptstyle\sum_{i=1}^{5} \left(2i-1\right)^2


[Expand]\scriptstyle\sum_{i=1}^{4} \left(4i\right)^2

Sum of Cubes

[Expand]\scriptstyle\sum_{i=1}^{5} i^3

[Expand]\scriptstyle\sum_{i=1}^{6} i^3

[Expand]\scriptstyle\sum_{i=1}^{7} i^3

[Expand]\scriptstyle\sum_{i=1}^{10} i^3

[Expand]\scriptstyle\sum_{i=4}^{10} i^3

[Expand]\scriptstyle\sum_{i=5}^{10} i^3

[Expand]\scriptstyle\sum_{i=5}^{12} i^3

Find the last term

[Expand]\scriptstyle\sum_{i=1}^{n} i^3=225

[Expand]\scriptstyle\sum_{i=1}^{n} i^3=3025

Cubes of Odds

[Expand]\scriptstyle\sum_{i=1}^{5} \left(2i-1\right)^3

[Expand]\scriptstyle\sum_{i=1}^{7} \left(2i-1\right)^3

[Expand]\scriptstyle\sum_{i=3}^{5} \left(2i-1\right)^3

[Expand]\scriptstyle\sum_{i=4}^{6} \left(2i-1\right)^3

Find the last term

[Expand]\scriptstyle\sum_{i=1}^{n} \left(2i-1\right)^3=256

[Expand]\scriptstyle\sum_{i=1}^{n} \left(2i-1\right)^3=1225

Cubes of Evens

[Expand]\scriptstyle\sum_{i=1}^{3} \left(2i\right)^3

[Expand]\scriptstyle\sum_{i=1}^{5} \left(2i\right)^3

[Expand]\scriptstyle\sum_{i=1}^{6} \left(2i\right)^3

[Expand]\scriptstyle\sum_{i=2}^{6} \left(2i\right)^3

[Expand]\scriptstyle\sum_{i=4}^{8} \left(2i\right)^3

Find the last term

[Expand]\scriptstyle\sum_{i=1}^{n} \left(2i\right)^3=400

[Expand]\scriptstyle\sum_{i=1}^{n} \left(2i\right)^3=1800


[Expand]\scriptstyle\sum_{i=1}^{5} \left(4i\right)^3

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