College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.5: Mathematical Induction
Problem 32E: Prove by induction that 1+2n3n for n1.
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Question
Let n ≥ 1 be an integer. Use induction to prove
2 + 4 + 6 + 8 + · · · + 2n = n(n + 1)
Expert Solution
Step 1
Given,
n ≥ 1 be an integer
We have to prove
2 + 4 + 6 + 8 + · · · + 2n = n(n + 1)
For n = 1
Left hand side = 2
Right hand side = 1(1+1) = 1 *2 = 2
Step 2
For n = k
Let's assume
2 + 4 + 6 + 8 + · · · + 2k = k(k + 1)
Now. To prove for n = k +1
Right hand side = (k+1)(k + 1+1) = (k+1)(k+2)
Left hand side = 2 + 4 + 6 + 8 + · · · + 2k + 2(k+1)
= k(k + 1) + 2(k+1) [Because 2 + 4 + 6 + 8 + · · · + 2k = k(k + 1) ]
= (k+1)(k+2)
= Right hand side
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