Let the coefficient of x r in the expansion of ( x + 3 ) n - 1 + ( x + 3 ) n - 2 ( x + 2 ) + ( x + 3 ) n - 3 ( x + 2 ) 2 + … + ( x + 2 ) n - 1 be r . If r = 0 nr =n -n , then the value of 2 +2 equals ______ . [2024]

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Solution

Q.
Let the coefficient of $x^{r}$ in the expansion of ${(x + 3)}^{n - 1} + {(x + 3)}^{n - 2} (x + 2) + {(x + 3)}^{n - 3} {(x + 2)}^{2} + \dots + {(x + 2)}^{n - 1}$ be $α_{r} .$ If $\sum_{r = 0}^{n} α_{r} = β^{n} - γ^{n}, β, γ \in N,$ then the value of $β^{2} + γ^{2}$ equals ______ . [2024]

Ans.
(25)

We have,

${(x + 3)}^{n - 1} + {(x + 3)}^{n - 2} (x + 2) + {(x + 3)}^{n - 3} {(x + 2)}^{2} + \dots + {(x + 2)}^{n - 1}$

$∴$ $\sum_{r = 0}^{n} α_{r} = 4^{n - 1} + 4^{n - 2} \times 3 + 4^{n - 3} \times 3^{2} + \dots + 3^{n - 1}$

$= 4^{n - 1} [1 + \frac{3}{4} + {(\frac{3}{4})}^{2} + \dots + {(\frac{3}{4})}^{n - 1}]$

$= 4^{n - 1} \times \frac{1 - {(\frac{3}{4})}^{n}}{1 - \frac{3}{4}} = 4^{n - 1} (1 - {(\frac{3}{4})}^{n}) (4)$

$= 4^{n} - 3^{n} = β - γ \Rightarrow b = 4, γ = 3$

$∴$ $β^{2} + γ^{2} = 16 + 9 = 25$

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