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

Jul 22, 2015

$- 510$

#### Explanation:

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

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

${S}_{20} = \left(\frac{20}{2}\right) \cdot \left(2 \cdot 22 + 19 \cdot \left(- 5\right)\right) = 10 \cdot \left(44 - 95\right)$

$= 10 \cdot \left(- 51\right) = - 510$.