Gujarat BoardEnglish MediumSTD 11 ScienceMATHSPRINCIPLE OF MATHEMATICAL INDUCTION1 Mark
MCQ
For all n∈N, 72n − 48n−1 is divisible by:
A
50
B
2304
✓
1234
D
44
✓
Answer
Correct option: C.
1234
2304
Solution: Concepts:
Suppose there is a given statement $\mathrm{P}(\mathrm{n})$ 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, $P(n)$ is true for all natural numbers $n$ Calculation:
Given:
$P(n)=72 n-48 n-1$
Put, $n=1$
$P(1)=72-48 \times 1-1=0$
Check the expression $\mathrm{P}(\mathrm{n})$ for $\mathrm{n}=\mathrm{k}$ (where k is some positive integer) $=2,3,4$.....
$P(2)=7^{2 n}-48 n-1=7^4-48 \times 2-1=2401-96-1=2401-97=2304$
$P(3)=7^{2 n}-48 n-1=7^6-48 \times 3-1=117649-144-1=117649-145=117504=2304 \times 51$
Since, all these numbers are divisible by 2304 for $\mathrm{n}=1$ and $\mathrm{k}=2,3, \ldots \ldots$
So, the given number is divisible by 2304
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