Find the sum of the arithmetic series

Question 1

Question 2

Answer:

132

Explanation:

Given arithmetic series:

$\displaystyle \sum^(12)_(k=1)(37-4k)$

$\textsf{Apply\:the\:Sum\:Rule}:\quad \displaystyle \sum a_n+b_n=\sum a_n+\sum b_n$

$\implies \displaystyle \sum^(12)_(k=1)37-\displaystyle \sum^(12)_(k=1)4k$

$\textsf{Apply\:the\:constant\:multiplication\:rule}:\quad \displaystyle \sum c\cdot a_n=c\cdot \sum a_n$

$\implies \displaystyle \sum^(12)_(k=1)37-\displaystyle 4\sum^(12)_(k=1)k$

Therefore:

$\implies 37 \cdot 12 - 4(1+2+3+4+5+6+7+8+9+10+11+12)$

$\implies 444 - 4(78)$

$\implies 444 - 312$

$\implies 132$

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