Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation Sk Sk-1 + 2k, for each integer k 21 S = 3 satisfies the following formula. Sn = 3 + n(n+1) for every integer n 20 Proof (by mathematical induction): Suppose So, S₁, S₂.... is a sequence that satisfies the recurrence relation Sk = Sk-1 + 2k for each integer k ≥ 1, with initial condition so = 3. Let the property P(n) be the equation s = 3+ n(n+1)✔ We will show that all the terms of the sequence satisfy the given explicit formula by showing that P(n) is true for every integer n 2 0. Show that P(0) is true: The left-hand side of P(0) is 50 ✓, which equals 3 ✓ .The right-hand side of P(0) is 3 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 ≥ 0, and suppose that P(k) is true. In other words, suppose that s = 3+k(k+1) ✓ Since the left-hand and right-hand sides equal each other, P(0) is true. [This is the inductive hypothesis.] Use the definition of So, S₁, S₂ to express Sk+ 1 in terms of S. and simplify the result using the inductive hypothesis. Your work should show that P(x + 1) is true, and therefore, that the sequence So, S₁, S₂ the given formula. defined by the given recurrence relation satisfies
Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation Sk Sk-1 + 2k, for each integer k 21 S = 3 satisfies the following formula. Sn = 3 + n(n+1) for every integer n 20 Proof (by mathematical induction): Suppose So, S₁, S₂.... is a sequence that satisfies the recurrence relation Sk = Sk-1 + 2k for each integer k ≥ 1, with initial condition so = 3. Let the property P(n) be the equation s = 3+ n(n+1)✔ We will show that all the terms of the sequence satisfy the given explicit formula by showing that P(n) is true for every integer n 2 0. Show that P(0) is true: The left-hand side of P(0) is 50 ✓, which equals 3 ✓ .The right-hand side of P(0) is 3 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 ≥ 0, and suppose that P(k) is true. In other words, suppose that s = 3+k(k+1) ✓ Since the left-hand and right-hand sides equal each other, P(0) is true. [This is the inductive hypothesis.] Use the definition of So, S₁, S₂ to express Sk+ 1 in terms of S. and simplify the result using the inductive hypothesis. Your work should show that P(x + 1) is true, and therefore, that the sequence So, S₁, S₂ the given formula. defined by the given recurrence relation satisfies
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Only need help with the final part. The blanks are already correct.
Specifically, Use the definition of s0, s1, s2, to express sk + 1 in terms of sk, and simplify the result using the inductive hypothesis.
![Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation
+ 2k, for each integer k≥1
Sk
So
= S
k - 1
= 3
satisfies the following formula.
Sn = 3 + n(n+1) for every integer n ≥ 0
Proof (by mathematical induction):
Suppose So, S₁, S2₁
Sk = Sk
is a sequence that satisfies the recurrence relation
k - 1 + 2k for each integer k ≥ 1, with initial condition so = 3.
Let the property P(n) be the equation =
Sn 3+ n(n+1)
Show that P(0) is true:
The left-hand side of P(0) is $0
which equals 3
Show that for each integer k ≥ 0, if P(k) is true, then P(k+ 1) is true:
[This is the inductive hypothesis.]
Use the definition of So, S₁, S2,
the given formula.
We will show that all the terms of the sequence satisfy the given explicit formula by showing that P(n) is true for every integer n ≥ 0.
Let k be any integer with k ≥ 0, and suppose that P(k) is true. In other words, suppose that sk = 3+k(k+ 1)
to express Sk + 1
. The right-hand side of P(0) is 3
. Since the left-hand and right-hand sides equal each other, P(0) is true.
in terms of Sk, and simplify the result using the inductive hypothesis. Your work should show that P(k + 1) is true, and therefore, that the sequence S0, S1, S2,
defined by the given recurrence relation satisfies](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb232dbf3-87c8-40c1-ad6d-cab85c52dec2%2F25834ab7-5d5e-4151-b18b-126c6bf7663f%2Fud1c6u_processed.png&w=3840&q=75)
Transcribed Image Text:Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation
+ 2k, for each integer k≥1
Sk
So
= S
k - 1
= 3
satisfies the following formula.
Sn = 3 + n(n+1) for every integer n ≥ 0
Proof (by mathematical induction):
Suppose So, S₁, S2₁
Sk = Sk
is a sequence that satisfies the recurrence relation
k - 1 + 2k for each integer k ≥ 1, with initial condition so = 3.
Let the property P(n) be the equation =
Sn 3+ n(n+1)
Show that P(0) is true:
The left-hand side of P(0) is $0
which equals 3
Show that for each integer k ≥ 0, if P(k) is true, then P(k+ 1) is true:
[This is the inductive hypothesis.]
Use the definition of So, S₁, S2,
the given formula.
We will show that all the terms of the sequence satisfy the given explicit formula by showing that P(n) is true for every integer n ≥ 0.
Let k be any integer with k ≥ 0, and suppose that P(k) is true. In other words, suppose that sk = 3+k(k+ 1)
to express Sk + 1
. The right-hand side of P(0) is 3
. Since the left-hand and right-hand sides equal each other, P(0) is true.
in terms of Sk, and simplify the result using the inductive hypothesis. Your work should show that P(k + 1) is true, and therefore, that the sequence S0, S1, S2,
defined by the given recurrence relation satisfies
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