How do you find the sum of the first 20 terms for: 22+17+12+7......?

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How do you find the sum of the first 20 terms for: 22+17+12+7......?

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Answer 1 · 2015-07-22T17:09:34

$- 510$ $-510$

Explanation:

You could first note that the last (20th) term added will be $22 - 5 \cdot 19 = - 73$ $22-5*19=-73$ (think about why this makes sense) and write your summation as $S = 22 + 17 + 12 + 7 + 2 + \dots + (- 63) + (- 68) + (- 73)$ $S=22+17+12+7+2+\cdots+(-63)+(-68)+(-73)$ . Now write $S = - 73 + (- 68) + (- 63) + (- 58) + (- 53) \dots + 12 + 17 + 22$ $S=-73+(-68)+(-63)+(-58)+(-53)\cdots+12+17+22$ directly beneath the first summation and add these equations to get $2 S = 20 \cdot (- 51) = - 1020$ $2S=20*(-51)=-1020$ (there are 20 terms that each add to $- 51$ $-51$ ). Now divide by 2 to get $S = - 510$ $S=-510$ .

Alternatively, you could recognize the summation as an arithmetic series with $n = 20$ $n=20$ terms, first term $a_{1} = 22$ $a_{1}=22$ , and "common difference" $d = 17 - 22 = - 5$ $d=17-22=-5$ . Then use the formula $S_{n} = (\frac{n}{2}) \cdot (2 a_{1} + (n - 1) d)$ $S_{n}=(n/2)*(2a_{1}+(n-1)d)$ to get

$S_{20} = (\frac{20}{2}) \cdot (2 \cdot 22 + 19 \cdot (- 5)) = 10 \cdot (44 - 95)$ $S_{20}=(20/2)*(2*22+19*(-5))=10*(44-95)$

$= 10 \cdot (- 51) = - 510$ $=10*(-51)=-510$ .

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How do you find the sum of the first 20 terms for: 22+17+12+7......?

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