Question

if 1 + 2 + 2 square + 2 Cube + 2 power 4 + ...........+n terms is 1025 then the value of n​

Answer 1

$1 + 2 + {2}^{2} + {2}^{3} + ... + n = 1025$

Lets compare first two terms,
2:1 = 2

then 2nd and third,
4:2=2

You'll notice that this ratio remains constant.

Therefore,
the terms are in GP aka geometric progression.

Now for sum till n terms in a GP is given by,

$a = (\frac{ {r}^{n} - 1}{r - 1} ) \: \: \: for \: r > 0$

where a is the sum and r is common ratio and n is the no. of terms,
Equating these two after subsituting the values,

$\frac{ {2}^{n} - 1 }{2 - 1} = 1025 \\ {2}^{n} - 1 = 1025 \\ {2}^{n} = 1024 \\ {2}^{n} = {2}^{10} \\ \\ n = 10$
Therefore we have n as 10.