Question

If first three terms in the expansion of a positive integral power of a binomial are 729, 7290 and 30375 respectively, find the binomial expansion.

​

Answer 1

SOLUTION

GIVEN

The first three terms in the expansion of a positive integral power of a binomial are 729, 7290 and 30375 respectively

TO DETERMINE

The binomial expansion.

EVALUATION

Let the binomial expansion is

$\sf{ {(a + b)}^{n} }$

$\sf{First \: Term \: \: {} = {}^{n} C_0 \: {a}^{n} = {a}^{n} }$

$\sf{Second \: Term \: \: {} = {}^{n} C_1 \: {a}^{n - 1}b = n {a}^{n - 1} b}$

$\displaystyle \: \sf{Third \: Term \: \: {} = {}^{n} C_2 \: {a}^{n - 2} {b}^{2} = \frac{n(n - 1)}{2} {a}^{n - 2} {b}^{2} }$

So by the given condition

$\displaystyle \: \sf{{a}^{n} = 729 \: \: \: .....(1) }$

$\displaystyle \: \sf{ n {a}^{n - 1} {b}^{} = 7290 \: \: \: \: \: \: ....(2)}$

$\displaystyle \: \sf{ \frac{n(n - 1)}{2} {a}^{n - 2} {b}^{2} = 30375 \: \: \: \: \: \: .....(3) }$

Equation (2) ÷ Equation (1) gives

$\displaystyle \: \sf{ \frac{nb}{a} = 10 } \: \: \: ......(4)$

Equation (3) ÷ Equation (2) gives

$\displaystyle \: \sf{ \frac{n - 1}{2} \times \frac{b}{a} = \frac{25}{6} \: \: \: \: \: \: \: \: ........(5) }$

From Equation (4) & Equation (5) we get

$\displaystyle \: \sf{ \frac{n - 1}{n} = \frac{5}{6} }$

$\displaystyle \: \sf{ \implies \: 6n - 6 = 5n}$

$\displaystyle \: \sf{ \implies \: n = 6}$

From Equation (1) we get

$\displaystyle \: \sf{ \implies \: {a}^{6} = 729}$

$\displaystyle \: \sf{ \implies \: {a}^{6} = {(3)}^{6} }$

$\displaystyle \: \sf{ \implies \: a = 3 }$

From Equation (4) we get

$\displaystyle \: \sf{ \implies \: \frac{6 \times b}{3} = 10 }$

$\displaystyle \: \sf{ \implies \: 2b = 10 }$

$\displaystyle \: \sf{ \implies \: b = 5 }$

Hence the required binomial expansion is

$\displaystyle \: \sf{ {(3 + 5)}^{6} }$

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. (32C0)^2-(32C1)^2+(32C2)^2-..........(32C32)^2=

https://brainly.in/question/28583245

2. IF 1/1!9!+1/3!7!+1/5!5!+1/7!3!+1/9!=2^n /10! then n=?

https://brainly.in/question/20427754