Name: Relation Between Binomial Coefficients
Uploaded: 2019-07-19
Duration: 3 min 15 s
Description: We have already studied the Binomial Coefficient and its properties. Let's watch this video to understand the relation between Binomial Coefficients.

Question

Using binomial theorem, prove that

6^{n} - 5 n - 1

is always divisible by

25,

where

n

is a natural number.

Accepted Answer

$6^{n} - 5 n - 1 = (1 + 5)^{n} - 5 n - 1$

$= (\begin{array}{l} n \\ 0 \end{array}) + (\begin{array}{l} n \\ 1 \end{array}) 5^{1} + (\begin{array}{l} n \\ 2 \end{array}) 5^{2} + (\begin{array}{l} n \\ 3 \end{array}) 5^{3} + \dots + (\begin{array}{l} n \\ n \end{array}) 5^{n} - 5 n - 1$

$= 1 + (\begin{array}{l} n \\ 1 \end{array}) 5^{1} + (\begin{array}{l} n \\ 2 \end{array}) 5^{2} + (\begin{array}{l} n \\ 3 \end{array}) 5^{3} + \dots + 5^{n} - 5 n - 1$

$= 1 + 5 n + 5^{2} ((\begin{array}{l} n \\ 2 \end{array}) + 5 (\begin{array}{l} n \\ 3 \end{array}) + \dots + 5^{n - 2}) - 5 n - 1$

$= 5^{2} ((\begin{array}{l} n \\ 2 \end{array}) + 5 (\begin{array}{l} n \\ 3 \end{array}) + \dots + 5^{n - 2})$

$= 25 \times$ Some natural number, $n \geq 2$

$∴ 6^{n} - 5 n - 1$ is divisible by $25$ for any natural number $n \geq 2$

Now, we have to check for $n = 1$

When $n = 1$

$6^{n} - 5 n - 1 = 6^{1} - 5 \times 1 - 1 = 6 - 5 - 1 = 0$

which is divisible by $25$ .

Hence, $6^{n} - 5 n - 1$ is divisible by $25$ for every natural number $n$

Using binomial theorem, prove that $6^{n} - 5 n - 1$ is always divisible by $25,$ where $n$ is a natural number.

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

Using binomial theorem, prove that 6n-5n-1 is always divisible by 25, where n is a natural number.

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

Using binomial theorem, prove that $6^{n} - 5 n - 1$ is always divisible by $25,$ where $n$ is a natural number.