Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# How Many Terms Are There in the A.P. Whose First and Fifth Terms Are −14 and 2 Respectively and the Sum of the Terms is 40? - Mathematics

How many terms are there in the A.P. whose first and fifth terms are −14 and 2 respectively and the sum of the terms is 40?

#### Solution

$\text { We have: }$

$a = - 14 \text { and } S_n = 40 . . . (i)$

$a_5 = 2$

$\Rightarrow a + \left( 5 - 1 \right)d = 2$

$\Rightarrow - 14 + 4d = 2$

$\Rightarrow 4d = 16$

$\Rightarrow d = 4 . . . (ii)$

$\text { Also }, S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]$

$\Rightarrow 40 = \frac{n}{2}\left[ 2\left( - 14 \right) + (n - 1) \times 4 \right] (\text { From }(i) \text { and } (ii))$

$\Rightarrow 80 = n\left[ - 28 + 4n - 4 \right]$

$\Rightarrow 80 = 4 n^2 - 32n$

$\Rightarrow n^2 - 8n - 20 = 0$

$\Rightarrow (n - 10)(n + 2) = 0$

$\Rightarrow n = 10, - 2$

$\text { But, n cannot be negative } .$

$\therefore n = 10$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 19 Arithmetic Progression
Exercise 19.4 | Q 16 | Page 31