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Notes

${\displaystyle \sum _{k=1}^{n}c=cn}$ (Sum of Constants Formula)

${\displaystyle \sum _{k=1}^{n}k={\displaystyle \frac{n(n+1)}{2}}}$ (Sum of Integers Formula)

${\displaystyle \sum _{k=1}^n{k}^2={\displaystyle \frac{n(n+1)(2n+1)}{6}}}$ (Sum of Squares Formula)

${\displaystyle \sum _{k=1}^n{k}^3={\displaystyle \frac{n^2(n+1{)}^2}{4}}}$ (Sum of Cubes Formula)

Practice Problems

$\mathbf{\text{1)}}$${\displaystyle \sum _{n=1}^8{n}^3}$

Show Answer

$1296$

Show Work

${\displaystyle \sum _{k=1}^n{k}^3={\displaystyle \frac{n^2(n+1{)}^2}{4}}}$

${\displaystyle \sum _{n=1}^8{n}^3={\displaystyle \frac{(8{)}^2((8)+1{)}^2}{4}}}$

${\displaystyle \sum _{n=1}^8{n}^3={\displaystyle \frac{64(9{)}^2}{4}}}$

${\displaystyle \sum _{n=1}^8{n}^3={\displaystyle \frac{5184}{4}}}$

${\displaystyle \sum _{n=1}^8{n}^3=1296}$

$\mathbf{\text{2)}}$${\displaystyle \sum _{n=1}^6{n}^2}$

Show Answer

$91$

Show Work

${\displaystyle \sum _{k=1}^n{k}^2={\displaystyle \frac{n(n+1)(2n+1)}{6}}}$

${\displaystyle \sum _{n=1}^6{n}^2={\displaystyle \frac{(6)((6)+1)(2(6)+1)}{6}}}$

${\displaystyle \sum _{n=1}^6{n}^2={\displaystyle \frac{(6)(7)(13)}{6}}}$

${\displaystyle \sum _{n=1}^6{n}^2=91}$

$\mathbf{\text{3)}}$${\displaystyle \sum _{n=1}^{4}7}$

Show Answer

$28$

Show Work

${\displaystyle \sum _{k=1}^{n}c=cn}$

${\displaystyle \sum _{n=1}^{4}7=(4)(7)}$

${\displaystyle \sum _{n=1}^{4}7=28}$

See Related Pages

$\bullet \text{ Arithmetic Sequences}$
$a_n=a_1+d(n-1)$

$\bullet \text{ Geometric Sequences}$
$a_n=a_1\cdot r^{(n-1)}\dots$

$\bullet \text{ Arithmetic Series}$
$s_n=\frac{n}{2}(a_1+a_n)\dots$

$\bullet \text{ Geometric Series}$
$s_n=a_1\frac{1-r^n}{1-r}\dots$

$\bullet \text{ Infinite Geometric Series}$
$s_{\mathrm{\infty }}=\frac{a_1}{1-r}|r|<1\dots$

$\bullet \text{ Summation Notation}$
${\displaystyle \sum _{i=4}^{9}3i-5\dots }$

$\bullet \text{ Recursive Sequences}$
$a_1=2,a_{n+1}=a_n+3\dots$

$\bullet \text{ Andymath Homepage}$

About Andymath.com

Andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. If you have any requests for additional content, please contact Andy at tutoring@andymath.com. He will promptly add the content.

Topics cover Elementary Math, Middle School, Algebra, Geometry, Algebra 2/Pre-calculus/Trig, Calculus and Probability/Statistics. In the future, I hope to add Physics and Linear Algebra content.

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