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S_(15) if
a_(1) and
a_(n)=-2a

$S_{15}$ if $a_{1}$ and $a_{n}=-2 a$

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Solve quadratic inequalities

Full solution

Q.

S_{15}

if

a_{1}

and

a_{n}=-2 a

Find common difference: Find the common difference $d$ of the arithmetic sequence. $a_n = a_1 + (n - 1)d$ $-2a = a_1 + (15 - 1)d$ $-2a = a_1 + 14d$ $d = \frac{-2a - a_1}{14}$
Calculate sum of first $15$ terms: Calculate the sum of the first $15$ terms $S_{15}$ using the formula for the sum of an arithmetic sequence. $\newline$ $S_{15} = \frac{n}{2} \cdot (a_{1} + a_{n})$ $\newline$ $S_{15} = \frac{15}{2} \cdot (a_{1} + (-2a))$ $\newline$ $S_{15} = \frac{15}{2} \cdot (a_{1} - 2a)$
Simplify expression for $S_{15}$ : Simplify the expression for $S_{15}$ .
$S_{15} = \frac{15}{2} \cdot (a_{1} + 2a)$
$S_{15} = \frac{15a_{1}}{2} - \frac{15 \cdot 2a}{2}$
$S_{15} = \frac{15a_{1}}{2} - 15a$

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