The value of remainder when 32 n -1-2 n -2- where n - 10 is divided by 7 isA- 1B- 2C- 0D- 5

The correct option is C 03^2n+1 + 2^n+2 ⇒ 3(9)^n + 4 (2)^n ⇒ 3(2 + 7)^n + 4 (2)^n ⇒ 3(^n C_0 7^0 2^n + ^n C_1 7^1 2^n 1 +^n C_1 7^2 2^n 2+ ⋯) nbsp; + 4 (2)^n ...

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Standard XII

Mathematics

Binomial Theorem for Any Index

The value of ...

Question

The value of remainder when $3^{2 n + 1} + 2^{n + 2}$ , where $n \geq 10$ is divided by $7$ is

1

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2

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0

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5

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Solution

The correct option is C $0$
$3^{2 n + 1} + 2^{n + 2} \Rightarrow 3 (9)^{n} + 4 (2)^{n} \Rightarrow 3 (2 + 7)^{n} + 4 (2)^{n} \Rightarrow 3 (^{n} C_{0} 7^{0} 2^{n} +^{n} C_{1} 7^{1} 2^{n - 1} +^{n} C_{1} 7^{2} 2^{n - 2} + \dots) + 4 (2)^{n} \Rightarrow 7 \cdot 2^{n} +^{n} C_{1} 7^{1} 2^{n - 1} +^{n} C_{1} 7^{2} 2^{n - 2} + \dots$

Hence the number is divisible by $7$ , so the remainder will be $0$ .

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The value of remainder when 32n+1+2n+2, where n≥10 is divided by 7 is

The correct option is C 032n+1+2n+2⇒3(9)n+4(2)n⇒3(2+7)n+4(2)n⇒3(nC0702n+nC1712n−1+nC1722n−2+⋯)+4(2)n⇒7⋅2n+nC1712n−1+nC1722n−2+⋯ Hence the number is divisible by 7, so the remainder will be 0.

The value of remainder when $3^{2 n + 1} + 2^{n + 2}$ , where $n \geq 10$ is divided by $7$ is