wiz-icon
MyQuestionIcon
MyQuestionIcon
20
You visited us 20 times! Enjoying our articles? Unlock Full Access!
Question

Prove the following by using the principle of mathematical induction for all nN:12+14+18+.....+12n=112n

Open in App
Solution

Let the given statement be P(n), i.e.,
P(n):12+14+18+.....+12n=112n
For n=1, we have
P(1)=12=1121=12, which is true.
Let P(k) be true for some kN, i.e.,
12+14+18+.....+12k=112k......(i)
We shall now prove that P(k+1) is true.
Consider (12+14+18+.....+12k)+12k+1
=(112k)+12k+1
=112k+12.2k
=112k(112)
=112k(12)
=112k+1
Thus P(k+1) is true whenever P(k) is true.
Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Relation between AM, GM and HM
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon