Observe that the successive squares differ by the successive odd integers. This
ID: 1720372 • Letter: O
Question
Observe that the successive squares differ by the successive odd integers. This can he expressed succinctly as n^2 = (2i - 1). Show that this is a consequence of the formula for k above. Now observe that there is a somewhat different pattern for the cubes: Explain why this pattern is equivalent to the formula k^3 = (2i - 1) and use the formulas that we gave previously for k^3 and k to verify this new formula. Observe that Find a formula that expresses the pattern we see here, and prove it. You may use the summation formulas above if you would like.Explanation / Answer
1.29)
formular
n^2 + (n^2 +1) +.....+ (n^2 + n) = (n^2 +n+1) + (n^2 +n+2) +.....+ (n^2 +n +n)
LHS = n^2(n+1) + n(n+1)/2 = n(n+1)[n +1/2] = n(n+1)(2n+1)/2
RHS = n(n^2+n) + n(n+1)/2 = n(n+1)[n +1/2] = n(n+1)(2n+1)/2
=>
LHS = RHS
thus proved
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