0.11 Let S(n) = 1 + 2 + + n be the sum of the first n natural numbers and let C(n) = 1³ +2³+...+n³ be the sum of the first n cubes. Prove the following equalities by induction on n, to arrive at the curious conclusion that C(n) = S² (n) for every n. a. S(n) = b. C(n)= n(n+1). (n + 2n³ + n²) = n²(n + 1)².
0.11 Let S(n) = 1 + 2 + + n be the sum of the first n natural numbers and let C(n) = 1³ +2³+...+n³ be the sum of the first n cubes. Prove the following equalities by induction on n, to arrive at the curious conclusion that C(n) = S² (n) for every n. a. S(n) = b. C(n)= n(n+1). (n + 2n³ + n²) = n²(n + 1)².
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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0.11 Let S(n) = 1 + 2 + + n be the sum of the first n natural numbers and let
C(n) = 1³ +23³ +...+n³ be the sum of the first n cubes. Prove the following
equalities by induction on n, to arrive at the curious conclusion that C(n) = S² (n)
3
for
every n.
a. S(n) =
b. C(n) =
n(n + 1).
(n + 2n³ + n²) = n²(n + 1)².
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