Can someone show a stepbystep of how to solve this Giventhat

Can someone show a step-by-step of how to solve this?

Giventhat(i+j(3+nry=9+84x+..., findthevalueof Given that I l +-x | (3 + nx)2-9 + 84x + … , find the value of n .

Solution

We have ( 1 + 2x/3)ⁿ ( 3 + nx)² = 9 + 84x + ....

or, [ (3 + 2x)ⁿ / 3ⁿ ] ( 9 + 6nx + n²x² ) = 9 + 84x + ....

or, [{ 3ⁿ + n* 3^n-1* 2x + ...+2ⁿxⁿ }/ 3ⁿ](9 + 6nx + n²x² ) = 9 + 84x +... ( all other terms in the 1st expression on the L.H.S. will contain varying powers of x other than 1)

or, ( 1 + 2nx/3 +...+ 2ⁿxⁿ /3ⁿ) ( 9 + 6nx + n²x²) = 9 + 84x +...

or, 9 + 6nx + 2n*9x /3 +.... = 9 + 84x + ...

or, 9 + 12nx +... = 9 + 84x + ...

Therefore, 12nx = 84 or, n = 84/ 12 = 7

NOTE:

We know that (a + b)² = a² + 2ab + b^{2 ;} (a + b)³ = a³ + 3a²b + 3ab² + b³ ; ( a + b)⁴ = a⁴ + 4a³ b + 6a² b² + 4ab³ + b⁴ and so on. On the same analogy, we have expanded ( a + b)ⁿ .