Question

Sum of the series is 31+29+27..... +7 by A.P

Answer 1

Answer:-

❀ Your Answer Is 228.

ExplanaTion:-

Given:-

• An AP.

• First term of the AP $\tt (a_1)$ = 31.

• Second term of the AP $\tt (a_2)$ = 29.

• Last term of the AP $\tt(a_n)$ = 7.

To Find:-

• The sum of all these terms of the AP.

Formula Used:-

$\\ \tt\longrightarrow S_n = \dfrac{n}{2}(a_1 + a_n).\\ \\ \tt\longrightarrow d = a_2-a_1.\\ \\ \tt\longrightarrow a_n = a+(n-1)d.$

Where

• n = Number of Terms.

• d = Common difference.

• $\tt S_n$ = Sum of n terms.

So Here,

$\\ \tt\longrightarrow d = a_2 - a_1. \\ \\ \tt\longrightarrow d = 29 - 31 . \\ \\ \longrightarrow { \boxed{ \underline{ \tt d = - 2.}}}$

So common difference is -2.

Also,

$\\ \tt\longrightarrow a_n = a + (n - 1)d. \\ \\ \tt\longrightarrow7 = 31 + (n - 1)( - 2). \\ \\ \tt\longrightarrow7 = 31 - 2n + 2. \\ \\ \tt\longrightarrow7 = 33 - 2n \\ \\ \tt\longrightarrow2n = 33 - 7. \\ \\ \tt\longrightarrow2n = 24. \\ \\ \tt\longrightarrow n = \cancel\frac{24}{2}. \\ \\ \tt\longrightarrow \boxed{ \underline{ \tt n = 12.}}$

So number of terms are 12.

And,

$\\ \tt\mapsto S_n = \frac{n}{2} (a + a_n). \\ \\ \tt\mapsto S_{12} = \cancel \frac{12}{2}(31 + 7). \\ \\ \tt\mapsto S_{12} = 6(38). \\ \\ \tt\mapsto { \boxed{ \underline{ \tt S_{12} = 228.}}}$

$\bf\therefore$ The sum of the series is 228.