Prove that displaystylesum_{j=1}^{n} 2^{j}=2^{n+1}-2 forallgeq1

Brittney Lord 2021-01-02 Answered
Prove that j=1n2j=2n+12
1
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Expert Answer

cheekabooy
Answered 2021-01-03 Author has 83 answers
Step 1
To proof: j=1n2j=2n+12 for all positive integers n.
Proof by induction
Let P(n) be the statement "j=1n2j=2n+12".
Basis step n=1
j=1n2j=j=112j=21=2
2n+12=21+12=222=2
Thus P(1) is true.
Inductive step
Let P(k) be true
j=1k2j=2k+12
We need to proof that P(k+1) is true
j=1k+12j
=[j=1k+12j]+2k+1
=2k+12+2k+1 Since P(k) is true
=2×2k+12 Combine loke terms
=2k+1+12
=2(k+1)+12
Thus P(k+1) is true
By the principle if mathematical induction, P(n) is true for all positive integers n.
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