P n - 52 n -1-3 n -2- 2 n -1 is divisible byA- 19B- 18C- 17D- 14

The correct option is A 19 P(n):5^2n+1+3^n+2.2^n 1 P(1): 5^3+3^3.1=152 →divisible by 19 Assume P(k) is true. 5^2k+1+3^k+2.2^k 1is divisible by 19 5^2k+1+3^k+2 ...

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Byju's Answer

Standard XI

Mathematics

Differentiation to solve modified sum of binomial coefficients

P n : 52 n +1...

Question

$P (n) : 5^{2 n + 1} + 3^{n + 2} {.2}^{n - 1}$ is divisible by

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Solution

The correct option is A
19

$P (n) : 5^{2 n + 1} + 3^{n + 2} {.2}^{n - 1}$
$P (1) : 5^{3} + 3^{3} .1 = 152 \to divisible by 19$
Assume P(k) is true.
$5^{2 k + 1} + 3^{k + 2} {.2}^{k - 1} is divisible by 19 5^{2 k + 1} + 3^{k + 2} {.2}^{k - 1} = 19 m P (k + 1) : 5^{2 k + 3} + 3^{k + 3} {.2}^{k} = {25.5}^{2 k + 1} + {6.3}^{k + 2} {.2}^{k - 1} = 6. (5^{2 k + 1} + 3^{k + 2} {.2}^{k - 1}) + {19.5}^{2 k + 1} = 19 (6 m + 5^{2 k + 1})$

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P(n):52n+1+3n+2.2n−1 is divisible by

The correct option is A 19 P(n):52n+1+3n+2.2n−1 P(1):53+33.1=152→divisible by 19 Assume P(k) is true. 52k+1+3k+2.2k−1is divisible by 1952k+1+3k+2.2k−1=19mP(k+1):52k+3+3k+3.2k=25.52k+1+6.3k+2.2k−1=6.(52k+1+3k+2.2k−1)+19.52k+1=19(6m+52k+1)

$P (n) : 5^{2 n + 1} + 3^{n + 2} {.2}^{n - 1}$ is divisible by