Gujarat BoardEnglish MediumSTD 11 ScienceMATHSPRINCIPLE OF MATHEMATICAL INDUCTION1 Mark
MCQ
For all $n \in N, 3.5^{2 n+1}+2^{3 n+1}$ is divisible by:
A
19
✓
17
C
23
D
25
✓
Answer
Correct option: B.
17
17
Solution:
Let $P(n)$ be the statement that $3.5^{2 n+1}+2^{3 n+1}$ is divisible by 17
If $\mathrm{n}=1$, then given expression $=3 \times 5^3+2^4+375+16=391=17 \times 23$, divisible by 17 .
$\mathrm{P}(1)$ is true
Assume that $P(k)$ is true.
$3.5^{2 \mathrm{k}+1}+2^{3 \mathrm{k}+1}$ is divisible by 17 .
$3 \cdot 5^{2 \mathrm{k}}=1+2^{3 \mathrm{k}+1}=17 \mathrm{~m}$ where $\mathrm{m} \in \mathrm{N}$
$3.5^{2(k+1)+1}+23^{(k+1)+1}$
$=3.5^{2 \mathrm{k}+1} \times 5^2+2^{3 \mathrm{k}+1} \times 2^3$
$=25^{(17 \mathrm{~m}-23 \mathrm{k}+1)}+8.2^{3 \mathrm{k}+1}$
$=425 \mathrm{~m}-25.2^{3 \mathrm{k}+1}+8.2^{3 \mathrm{k}+1}$
$=425 \mathrm{~m}-17.2^{3 \mathrm{k}+1}$
$=17\left(25 m-2^{3 k+1}\right)$, divisible by 17
$\mathrm{P}(\mathrm{k}+1)$ is true by Principle of Mathematical Induction
$P(n)$ is true for all $n \in N .3 .5^{2 n+1}+2^{3 n+1}$ is divisible by 17 for all $n \in N$
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