Gujarat BoardEnglish MediumSTD 11 ScienceMATHSPRINCIPLE OF MATHEMATICAL INDUCTION1 Mark
MCQ
For every positive integer $n, 7n – 3n$ is divisible by
A
2
✓
4
C
5
D
6
✓
Answer
Correct option: B.
4
4
Solution: Concept:
Suppose there is a given statement $P(n)$ involving the natural number $n$ such that
- The statement is true for $n=1$, i.e., $P(1)$ is true, and
- If the statement is true for $n=k$ (where $k$ is some positive integer), then the statement is also true for $n=k+$ 1, i.e., truth of $P(k)$ implies the truth of $P(k+1)$.
Then, $\mathrm{P}(\mathrm{n})$ is true for all natural numbers n Calculation:
We have to find $7^n-3^n$ is divisible by which number
Consider $\mathrm{P}(\mathrm{n})$ : $7 \mathrm{n}-3 \mathrm{n}$
$P(1): 7^1-3^1=4$
Thus, $7 n-3 n$ is divisible by 4
Let $P(k)$ is true for $n=K$
$\Rightarrow 7^{\mathrm{k}}-3^{\mathrm{k}}$ is divisible by 4
So, $7 n-3 n=4 d$
Now, prove that $\mathrm{P}(\mathrm{k}+1)$ is true.
$\Rightarrow 7^{(k+1)}-3^{(k+1)}=7^{(k+1)}-7.3^k+7.3^k-3^{(k+1)}$
$=7\left(7^k-3^k\right)+(7-3) 3^k$
$=7(4 d)+(7-3) 3^k$
$=7(4 d)+4.3^k$
$=4\left(7 \mathrm{~d}+3^k\right)$
Hence, $P(n): 7^n-3^n$ is divisible by 4 is true.
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