Question

arithmetic progression
if in an ap the 10th term of 10 times is equal to 15th term of 15 time, then the find of 25th term?

Answer 1

10 t 10 = 15 t 15

tn = a+(n-1)d

10(a+(10-1)d) = 15(a+(15-1)d)

10 a +90 d  = 15 a + 210 d

d(90 - 210) = a(15 - 10)

-120 d  = 5 a

-120     =   a

   5           d

a / d   = -24 / 1

so, a=-24 & d = 1

then t 25   = a + (n-1) d

                = -24 +(25-1) 1

                = -24 + 24

                = 0

Answer 2

Hello!

$\star \: \sf 10th\: term\: of \:A.P. = \sf a+9d \\ \star \: \sf 15th\: term \:of\: A.P. = \sf a+14d$

ATQ,

$\sf 10(a+9d) = \sf 15(a+14d) \\ \\ \implies \sf 10a+90d = \sf 15a+210d \\ \\ \implies \sf 90d-210d = \sf 15a-10a \\ \\ \implies \sf 5a = \sf -120d \\ \\ \implies \sf a = - \dfrac{120}{5} \sf d \\ \\ \implies \sf a = \sf -24d$

$\underline{\bf{25th\:term}}$ :
 
$\sf a_{25} = \sf a+24d \\ \\ \implies \sf -24d + 24d \implies \boxed{\red{\bf{0}}}$

Hence,
The 25th term of the A.P. is $\bf{0}$

Cheers!