# How Many Terms of the A.P. −6, − 11 2 , −5, ... Are Needed to Give the Sum −25? - Mathematics

How many terms of the A.P. −6, $- \frac{11}{2}$, −5, ... are needed to give the sum −25?

#### Solution

$\text { Given: }$

$\text{ An } \hspace{0.167em} A . P .\text { with a = - 6 and d }= - \frac{11}{2} - \left( - 6 \right) = \frac{1}{2}$

$S_n = - 25$

$\therefore - 25 = \frac{n}{2}\left[ 2 \times \left( - 6 \right) + \left( n - 1 \right)\frac{1}{2} \right]$

$\Rightarrow - 25 = \frac{n}{2}\left[ - 12 + \frac{n}{2} - \frac{1}{2} \right]$

$\Rightarrow - 50 = n\left[ \frac{n}{2} - \frac{25}{2} \right]$

$\Rightarrow - 100 = n\left( n - 25 \right)$

$\Rightarrow n^2 - 25n + 100 = 0$

$\Rightarrow \left( n - 20 \right)\left( n - 5 \right) = 0$

$\Rightarrow n = 20 \text { or } n = 5$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 19 Arithmetic Progression
Exercise 19.4 | Q 28 | Page 31