Question

Find the sum of the arithmetic series

Answer 1

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`