If n is a positive integer, prove $3^{3 n} - 26 n - 1$ is divisible by 676.

Open in App

Solution

$3^{3 n} - 26 n - 1$

$= (3^{3})^{n} - 26 n - 1 = 27^{n} - 26 n - 1$

=(1+26)n-26n-1

$= (^{n} C_{0} +^{n} C_{1} (26)^{1} +^{n} C_{2} (26)^{2} + . . . . . +^{n} C_{n} (26)^{n}) - 26 n - 1$

$= (1 + 26 n + 676^{n} C_{2} + . . . . + 676 (26)^{n - 2} - 26 n - 1)$

$= 676 (^{n} C_{2} + . . . . + (26)^{n - 2})$

$∴ 3^{3 n} - 26 n - 1$ is divisible by 676 for $n \in N .$

Hence, proved.

Suggest Corrections

Similar questions

3^{3 n} - 26 n - 1

676

\in

\forall n \in N, 3^{3 n} - 26^{n}

If r is a fixed positive integer, prove by induction that (r+1)(r+2)(r+3)....(r+n) is divisible by n!

1^{n} + 8^{n} - 3^{n} - 6^{n}

p^{n + 1} + (p + 1)^{2 n - 1}

p^{2}

Join BYJU'S Learning Program