To show that Pk+1 is true, add (2(k+ 1))3 to both sides of Pk. +(2k)³ +(2k +1 23 +43 +63 + ... Completely factor the right-hand side. 23 +43 +63 +...+(2(k+ 1))³ = 2(k+1 = 2(k+1 = 2(k+1 Rewrite the right-hand side in the desired form. ))³ = 2k²(k + 1)² + (2【 k +1 1)²+(2(k+1,)) ],» )²(4(k+1) )²(k² + 4k+ 4 )²(x + 2 23 + 4³ + 6³ + ... + (2(k + 1))³ = 2 (k+ 1)² ( (k+1) + 1)² X + 4( k + 1 2+ ) 1,9 3² ))

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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To show that Pk+1 is true, add (2(k+ 1))3 to both sides of Pk.
+(2k)³ +(2k +1
23 +4³ +6³ +
+
6³ + ... + (2k)³ +
Completely factor the right-hand side.
23 + 43 + 6³ + ... + (2(k + 1))³ = 2( k +1
2k+1
2 k +1
Rewrite the right-hand side in the desired form.
3
))³ = 2k²(k + 1)² + (2k + 1
)²( 4 (k+1)
) ² (k² + 4k +4
)²(x + 2
23 + 4³ + 6³ + ... + (2(k + 1))³ = 2 (k+ 1)² ( (k+ 1) + 1)²
X
2+
4k+1
)
. ))³
2
)²
Transcribed Image Text:To show that Pk+1 is true, add (2(k+ 1))3 to both sides of Pk. +(2k)³ +(2k +1 23 +4³ +6³ + + 6³ + ... + (2k)³ + Completely factor the right-hand side. 23 + 43 + 6³ + ... + (2(k + 1))³ = 2( k +1 2k+1 2 k +1 Rewrite the right-hand side in the desired form. 3 ))³ = 2k²(k + 1)² + (2k + 1 )²( 4 (k+1) ) ² (k² + 4k +4 )²(x + 2 23 + 4³ + 6³ + ... + (2(k + 1))³ = 2 (k+ 1)² ( (k+ 1) + 1)² X 2+ 4k+1 ) . ))³ 2 )²
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