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Worked solution

Count repetitions in a constant series

Evaluate (a) ∑r=−275\sum_{r=-2}^{7}5; (b) ∑r=310(−2)\sum_{r=3}^{10}(-2).

CJC · Sequences and Series · Example 18

(a) Ten positive constants

Answer

5050.

Count all integer indices, including the negative ones

∑r=−275=5+5+5+⋯+5=[7−(−2)+1](5)=10(5)=50.\begin{aligned}\sum_{r=-2}^{7}5&=5+5+5+\cdots+5\\&=[7-(-2)+1](5)\\&=10(5)=50.\end{aligned}

The constant 5 is repeated once per index.

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(b) Eight negative constants

Answer

−16-16.

Multiply the repeated constant by the term count

∑r=310(−2)=(−2)+(−2)+(−2)+⋯+(−2)=(10−3+1)(−2)=8(−2)=−16.\begin{aligned}\sum_{r=3}^{10}(-2)&=(-2)+(-2)+(-2)+\cdots+(-2)\\&=(10-3+1)(-2)\\&=8(-2)=-16.\end{aligned}

Multiply eight copies of −2; keep the negative constant in brackets.

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