The sum, $$\sum\limits_{n = 1}^7 {{{n\left( {n + 1} \right)\left( {2n + 1} \right)} \over 4}} $$ is equal to ________.
Answer (integer)
504
Solution
$$\sum\limits_{n = 1}^7 {{{n\left( {n + 1} \right)\left( {2n + 1} \right)} \over 4}} $$
<br><br>= ${1 \over 4}\sum\limits_{n = 1}^7 {\left( {2{n^3} + 3{n^2} + n} \right)}$
<br><br>= ${1 \over 2}\sum\limits_{n = 1}^7 {{n^3}}$ + ${3 \over 4}\sum\limits_{n = 1}^7 {{n^2}}$ + ${1 \over 4}\sum\limits_{n = 1}^7 n$
<br><br>= ${1 \over 2}{\left( {{{7\left( {7 + 1} \right)} \over 2}} \right)^2}$ + $${3 \over 4}\left( {{{7\left( {7 + 1} \right)\left( {14 + 1} \right)} \over 6}} \right)$$ + ${1 \over 4}{{7\left( 8 \right)} \over 2}$
<br><br>= (49)(8) + (15$\times$7) + (7)
<br><br>= 392 + 105 + 7 = 504
About this question
Subject: Mathematics · Chapter: Sequences and Series · Topic: Arithmetic Progression
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