# If 9th Term of an A.P. is Zero, Prove that Its 29th Term is Double the 19th Term. - Mathematics

If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.

#### Solution

Given:

$a_9 = 0$

$\Rightarrow a + \left( 9 - 1 \right)d = 0 \left[ a_n = a + \left( n - 1 \right)d \right]$

$\Rightarrow a + 8d = 0$

$\Rightarrow a = - 8d . . . (i)$

To prove:

$a_{29} = 2 a_{19}$

Proof:

$\text { LHS }: a_{29} = a + \left( 29 - 1 \right)d$

$= a + 28d$

$= - 8d + 28d \left( \text { From }(i) \right)$

$= 20d$

$RHS: 2 a_{19} = 2\left[ a + \left( 19 - 1 \right)d \right]$

$= 2(a + 18d)$

$= 2a + 36d$

$= 2( - 8d) + 36d \left( \text { From }(i) \right)$

$= - 16d + 36d$

$= 20d$

LHS = RHS Hence, proved.

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 19 Arithmetic Progression
Exercise 19.2 | Q 9 | Page 12