Put the following statements into order to prove the summation formula P(n) = (Co² = (+1)(2+1) ) for all nЄ N. Put N next to the statements that should not be used. Note that N = {1,2,3,...) is the set of positive integers. 1. Now suppose we have proved the statement P(n) for some nЄ N. 2. We simplify the right side: (+1)(n+1)+(n+1)²= (n+1)(n+1) +n+1)= 3. The statement is true for n = 1 because in that case, both sides of the equation are 1. 4. Thus, we have proved the statement P(n+1)=((+1)(n+2)(2n+3)) 5. By adding n + 1 to both sides of P(n), we get 6. Now suppose we have proved the statement P(n) for all nЄN. 7. Σ + (η + 1) Σ n(n+1)(2n+1) +(n+1)³ 8. By adding (n+1)2 to both sides of P(n), we get 9. The statement is true for n = 1 because in that case, both sides of the equation are 0. 10. Σ 1+n+1= Σ +1 (n+1)(n+2)(2n+3) (+1)(2+1) +n+1 6 11. We simplify the right side: (n+1)(2+1) +(n+1)=(n+1)((+1)+1)=(+1)(+2)(2+3)

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 44RE
Question

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Put the following statements into order to prove the summation formula P(n) = (Co² = (+1)(2+1) ) for all nЄ N. Put N next to the statements
that should not be used. Note that N = {1,2,3,...) is the set of positive integers.
1. Now suppose we have proved the statement P(n) for some nЄ N.
2. We simplify the right side: (+1)(n+1)+(n+1)²= (n+1)(n+1)
+n+1)=
3. The statement is true for n = 1 because in that case, both sides of the equation are 1.
4. Thus, we have proved the statement P(n+1)=((+1)(n+2)(2n+3))
5. By adding n + 1 to both sides of P(n), we get
6. Now suppose we have proved the statement P(n) for all nЄN.
7. Σ
+ (η + 1) Σ
n(n+1)(2n+1)
+(n+1)³
8. By adding (n+1)2 to both sides of P(n), we get
9. The statement is true for n = 1 because in that case, both sides of the equation are 0.
10. Σ 1+n+1= Σ
+1
(n+1)(n+2)(2n+3)
(+1)(2+1)
+n+1
6
11. We simplify the right side:
(n+1)(2+1)
+(n+1)=(n+1)((+1)+1)=(+1)(+2)(2+3)
Transcribed Image Text:Put the following statements into order to prove the summation formula P(n) = (Co² = (+1)(2+1) ) for all nЄ N. Put N next to the statements that should not be used. Note that N = {1,2,3,...) is the set of positive integers. 1. Now suppose we have proved the statement P(n) for some nЄ N. 2. We simplify the right side: (+1)(n+1)+(n+1)²= (n+1)(n+1) +n+1)= 3. The statement is true for n = 1 because in that case, both sides of the equation are 1. 4. Thus, we have proved the statement P(n+1)=((+1)(n+2)(2n+3)) 5. By adding n + 1 to both sides of P(n), we get 6. Now suppose we have proved the statement P(n) for all nЄN. 7. Σ + (η + 1) Σ n(n+1)(2n+1) +(n+1)³ 8. By adding (n+1)2 to both sides of P(n), we get 9. The statement is true for n = 1 because in that case, both sides of the equation are 0. 10. Σ 1+n+1= Σ +1 (n+1)(n+2)(2n+3) (+1)(2+1) +n+1 6 11. We simplify the right side: (n+1)(2+1) +(n+1)=(n+1)((+1)+1)=(+1)(+2)(2+3)
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