My first problem!

Algebra Level 4

$1 \cdot 4 + 4 \cdot 7 + 7 \cdot10 + 10 \cdot13 + \cdots + 97\cdot100$

Find the sum of digits of the value of the sum above.

For example, if the sum is 12, then submit your answer as $1+2=3$ .

The answer is 6.

1 solution

Mark Recio
Jul 9, 2016

The $k^{th}$ term is $a_k= (3k-2)(3k+1)=9k^2-3k-2$ . Thus, there are $n=33$ terms.

$\sum_{n=1}^{n } a_k$ = $9* \sum_{k=1}^{n } k^2$ $-3*\sum_{k=1}^{n } k$ $-2n$

$=9* \frac{n(n+1)(2n+1)}{6}$ ) $-3* \frac{n(n+1)}{2}$ $-2n$

= $n(3n^2+3n-2)$

Therefore, substituting $n=33$ , $33[3(33)^2+3(33)-2]=111012$

Thus, $1+1+1+0+1+2=6$

Use " \cdot " Or " \times " for the symbol of multiplication and " \left( ... \right) " to adjust the size of brackets. " \quad " is used to give space between characters . A Whole bunch of other symbols here ... Keep posting :)

Sabhrant Sachan - 4 years, 11 months ago

Typo: $\displaystyle \sum_{\color{#3D99F6}{k=1}}^n a_k$

Hung Woei Neoh - 4 years, 11 months ago

Same solution :)

Eric Escober - 4 years, 11 months ago

My first problem!

The answer is 6.

1 solution

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