Using binomial theorem, prove that (2^ 3 n -7 n -1 ) is divisible by 49 , where n N

To prove: (2^ 3 n -7 n-1 ) is divisible by 49 , where n N l (2^ 3 n -7 n-1 )= (2^3 ) n-7 n-1 =8^n-7 n-1 =(1+7)^n-7 n-1 Now using binomial theorem.. l ^n C_0 1^n+ ^n C_1 1^ n-1 7+ ^n C_2 1^ n-2 7^2+ + ^n C_ n-1 7^ n-1 + ^n C_n 7^n-7 n-1 = ^n C_0+ ^n C_1 7+ ^n C_2 7^2+ + ^n C_ n-1 7^ n-1 + ^n C_n 7^n-7 n-1 =1+7 n+7^2 [ ^n C_2+ ^n C_3 7+ + ^n C_ n-1 7^ n-3 + ^n C_n 7^ n-2 ]-7 n-1 =7^2 [ ^n C_2+ ^n C_3 7+ + ^n C_ n-1 7^ n-3 + ^n C_n 7^ n-2 ] =49 [ ^n C_2+ ^n C_3 7+ + ^n C_ n-1 7^ n-3 + ^n C_n 7^ n-2 ] =49 K, where K= ( ^n C_2+ ^n C_3 7+ ++ ^n C_ n-1 7^ n-3 + ^n C_n 7^ n-2 ) Now, (2^ 3 n -7 n-1 )=49 K Therefore (2^ 3 n -7 n-1 ) is divisible by 49 .

Using binomial theorem, prove that (2^ 3 n -7 n -1 ) is divisible…

Download App

Question

Using binomial theorem, prove that $\left(2^{3 n }-7 n -1\right)$ is divisible by 49 , where $n \in N$

✓

Answer

To prove: $\left(2^{3 n}-7 n-1\right)$ is divisible by 49 , where $n \in N$
$\begin{array}{l}\left(2^{3 n}-7 n-1\right)=\left(2^3\right) n-7 n-1 \\ =8^n-7 n-1 \\ =(1+7)^n-7 n-1\end{array}$
Now using binomial theorem..
$\begin{array}{l}\Rightarrow{ }^n C_0 1^n+{ }^n C_1 1^{n-1} 7+{ }^n C_2 1^{n-2} 7^2+\ldots \ldots+{ }^n C_{n-1} 7^{n-1}+{ }^n C_n 7^n-7 n-1 \\ ={ }^n C_0+{ }^n C_1 7+{ }^n C_2 7^2+\ldots \ldots+{ }^n C_{n-1} 7^{n-1}+{ }^n C_n 7^n-7 n-1 \\ =1+7 n+7^2\left[{ }^n C_2+{ }^n C_3 7+\ldots+{ }^n C_{n-1} 7^{n-3}+{ }^n C_n 7^{n-2}\right]-7 n-1 \\ =7^2\left[{ }^n C_2+{ }^n C_3 7+\ldots+{ }^n C_{n-1} 7^{n-3}+{ }^n C_n 7^{n-2}\right] \\ =49\left[{ }^n C_2+{ }^n C_3 7+\ldots+{ }^n C_{n-1} 7^{n-3}+{ }^n C_n 7^{n-2}\right] \\ =49 K, \text { where } K=\left({ }^n C_2+{ }^n C_3 7+\ldots++{ }^n C_{n-1} 7^{n-3}+{ }^n C_n 7^{n-2}\right)\end{array}$
Now, $\left(2^{3 n}-7 n-1\right)=49 K$
Therefore $\left(2^{3 n}-7 n-1\right)$ is divisible by 49 .

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Model Paper 3→Model Paper 3 — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Model Paper 3 — MATHS STD 11 Science — Question

Answer

Need a full question paper?

Explore more

Similar questions