Prove the following statement by mathematical induction.n+1For every integer n 2 0, ri:2' = n · 2" + 2 + 2.i = 1Proof (by mathematical induction): Let P(n) be the equationi. 2' = n. 2" + 2 + 2.i = 1We will show that P(n) is true for every integer n 2 0.Show that P(0) is true: Select P(0) from the choices below.0+1Fi. 2' = 0 - 2° + 2 + 2i= 1O 2 = 0 . 27 + 2 + 2F1.2' = 1 - 21 + 2 +i = 0i: 2' = 0. 20 + 2 + 2The selected statement is true because both sides of the equation equal the same quantity.Show that for each integer k > 0, if P(k) is true, then P(k + 1) is true:Let k be any integer with k 2 0, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choices below.k+1i = 1k+1i. 2kk 2i. 2k +The right-hand side of P(k) is 1[The inductive hypothesis states that the two sides of P(k) are equal.](k+1)+1We must show that P(k + 1) is true. The left-hand side of P(k + 1) is S(k+1)+1(k+1)+1i. 2'. When the final term ofi- 2' is written separately, the result isi- 2' =i: 2' +. The right-hand side of P(k + 1) isAfter substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomesi = 1i = 1i = 1i = 1+ (k:+ 2)2k + 2 ). when the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal+ 2. Hence P(k + 1) is true, which completes the inductive step.[Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]

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Description: Prove the following statement by mathematical induction.n+1For every integer n 2 0, ri:2' = n · 2" + 2 + 2.i = 1Proof (by mathematical induction): Let P(n) be the equationi. 2' = n. 2" + 2 + 2.i = 1We will show that P(n) is true for every integer n

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Prove the following statement by mathematical induction. For every integer na 0, 52n2*.2. Proof (by mathematical induction): Let P(n) be the equation We will show that P(n) is true for every integer n20. Show that P(0) is true: Select P(0) from the choices below. 02.0.2*2. 2 lo1.2 2. The selected statement is true because both sides of the equation equal the same quantity. Show that for each integer k0, i P(A) is true, then P(k + 1) is true Let k be any integer with k 20, and suppose that PK) is true. Select the expression for the left-hand side of P(K) from the choices below.

Prove the following statement by mathematical induction. For every integer na 0, 52n2.2. Proof (by mathematical induction): Let P(n) be the equation We will show that P(n) is true for every integer n20. Show that P(0) is true: Select P(0) from the choices below. 02.0.22. 2 lo1.2 2. The selected statement is true because both sides of the equation equal the same quantity. Show that for each integer k0, i P(A) is true, then P(k + 1) is true Let k be any integer with k 20, and suppose that PK) is true. Select the expression for the left-hand side of P(K) from the choices below.

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

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Question

Prove the following statement by mathematical induction.
n+1
For every integer n 2 0, ri:2' = n · 2" + 2 + 2.
i = 1
Proof (by mathematical induction): Let P(n) be the equation
i. 2' = n. 2" + 2 + 2.
i = 1
We will show that P(n) is true for every integer n 2 0.
Show that P(0) is true: Select P(0) from the choices below.
0+1
Fi. 2' = 0 - 2° + 2 + 2
i= 1
O 2 = 0 . 27 + 2 + 2
F1.2' = 1 - 21 + 2 +
i = 0
i: 2' = 0. 20 + 2 + 2
The selected statement is true because both sides of the equation equal the same quantity.
Show that for each integer k > 0, if P(k) is true, then P(k + 1) is true:
Let k be any integer with k 2 0, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choices below.
k+1
i = 1
k+1
i. 2k
k 2
i. 2k +
The right-hand side of P(k) is 1
[The inductive hypothesis states that the two sides of P(k) are equal.]
(k+1)+1
We must show that P(k + 1) is true. The left-hand side of P(k + 1) is S
(k+1)+1
(k+1)+1
i. 2'. When the final term of
i- 2' is written separately, the result is
i- 2' =
i: 2' +
. The right-hand side of P(k + 1) is
After substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomes
i = 1
i = 1
i = 1
i = 1
+ (k
:+ 2)2k + 2 ). when the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal
+ 2. Hence P(k + 1) is true, which completes the inductive step.
[Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]

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