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

# In a Certain A.P. the 24th Term is Twice the 10th Term. Prove that the 72nd Term is Twice the 34th Term. - Mathematics

In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.

#### Solution

Given:

$a_{24} = 2 a_{10}$

$\Rightarrow a + \left( 24 - 1 \right)d = 2\left[ a + \left( 10 - 1 \right)d \right]$

$\Rightarrow a + 23d = 2(a + 9d)$

$\Rightarrow a + 23d = 2a + 18d$

$\Rightarrow 5d = a . . . (i)$

$\text { To prove }:$

$a_{72} = 2 a_{34}$

$\text { LHS: } a_{72} = a + \left( 72 - 1 \right)d$

$\Rightarrow a_{72} = a + 71d$

$\Rightarrow a_{72} = 5d + 71d \left( \text { From }(i) \right)$

$\Rightarrow a_{72} = 76d$

$\text { RHS }: 2 a_{34} = 2\left[ a + \left( 34 - 1 \right)d \right]$

$\Rightarrow 2 a_{34} = 2\left( a + 33d \right)$

$\Rightarrow 2 a_{34} = 2(5d + 33d) \left( \text { Form }(i) \right)$

$\Rightarrow 2 a_{34} = 2\left( 38d \right)$

$\Rightarrow 2 a_{34} = 76d$

∴ RHS = LHS
Hence, proved.

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 19 Arithmetic Progression
Exercise 19.2 | Q 12 | Page 12