# If 10 Times the 10th Term of an A.P. is Equal to 15 Times the 15th Term, Show that 25th Term of the A.P. is Zero. - Mathematics

If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.

#### Solution

Given:

$10 a_{10} = 15 a_{15}$

$\Rightarrow 10\left[ a + \left( 10 - 1 \right)d \right] = 15\left[ a + \left( 15 - 1 \right)d \right]$

$\Rightarrow 10(a + 9d) = 15(a + 14d)$

$\Rightarrow 10a + 90d = 15a + 210d$

$\Rightarrow 0 = 5a + 120d$

$\Rightarrow 0 = a + 24d$

$\Rightarrow a = - 24d . . . (i)$

To show:

$a_{25} = 0$

$\Rightarrow \text { LHS }: a_{25} = a + \left( 25 - 1 \right)d$

$= a + 24d$

$= - 24d + 24d \left( \text { From }(i) \right)$

$= 0 = \text { RHS }$

$\text { Hence, proved } .$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 19 Arithmetic Progression
Exercise 19.2 | Q 10 | Page 12