Prove by mathematical induction method : [4]
34n+2 + 52n+1 is multiple of 14.
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11th
Maths
Principle of Mathematical Induction
Introduction to Mathematical Induction
For all n ∈ N , 10^n + 3.4^...
MATHS
For all n∈N, 10
n
+3.4
n+2
+5 is divisible by
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ANSWER
10
n
+3.4
n+2
+5
Put n=1
10+3(64)+5=207 which is divisible by 3,9,23,207
We will check for n=2
100+768+5=873
which is divisible by 3 and 9 only.
Every number divisible by 9 is also divisible by 3.
Now, let P(n) is true
i.e. 10
n
+3.4
n+2
+5 is divisible by 9
⇒10
n
+3.4
n+2
+5=9λ ....(1)
Now, we will check for P(n+1)
Consider, 10
n+1
+3.4
n+3
+5
=10
n
10+3.4
n+2
4+5
=10(9λ−3.4
n+2
−5)+3.4
n+2
4+5 (by (1))
=90λ−27.4
n+2
−45
=9(10λ−3.4
n+2
−5)
Answer:
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