Question:
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:
$10 a_{10}=15 a_{15}$
$\Rightarrow 10[a+(10-1) d]=15[a+(15-1) d]$
$\Rightarrow 10(a+9 d)=15(a+14 d)$
$\Rightarrow 10 a+90 d=15 a+210 d$
$\Rightarrow 0=5 a+120 d$
$\Rightarrow 0$
$\Rightarrow 0=a+24 d$
$\Rightarrow a=-24 d \ldots(\mathrm{i})$
To show:
$a_{25}=0$
$\Rightarrow$ LHS : $a_{25}=a+(25-1) d$
$=a+24 d$
$=-24 d+24 d \quad(\operatorname{From}(\mathrm{i}))$
$=0=\mathrm{RHS}$
Hence proved.
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