Zadanie 1c
Predict the output at input 25, to the nearest integer.
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Znaleziono 1343 zadania
Predict the output at input 25, to the nearest integer.
Show that S1(n) = n2.
For n=1,2,3, the values of S2(n) are 1,10,p. (i) State p. (ii) Suppose S2(n) = a1n + a2n2 + a3n3 for rational coefficients a₁,a₂,a₃. Write three linear equations for these coefficients. (iii) Hence determine the coefficients.
Let F(x) = x + x3 + x5 + ⋯ + x2n−1. Show that .
Define and Fq(x) = xFq−1′(x) for q≥1. (i) Show that . (ii) Prove by induction on q that for q≥1. (iii) Write F_q(1) in sigma notation.
For F(x) = x + x3 + ⋯ + x2n−1, use a geometric sum to show that F(x) = x(1 − x2n)/(1 − x2) whenever x≠±1.
Let F(x) = x(1 − x2n)/(1 − x2) for x≠±1 and F1 = xF′. Show that F1(x) = [x + x3 − (2n + 1)x2n+1 + (2n − 1)x2n+3]/(1 − x2)2.
For x≠±1 let F1(x) = [x + x3 − (2n + 1)x2n+1 + (2n − 1)x2n+3]/(1 − x2)2. (i) Show that direct substitution into gives an indeterminate form. (ii) Use l’Hôpital’s rule to prove that the limit is n².
Find (i) the centre of the lower base; (ii) its radius.
Find the exact volume of material remaining after the cone is removed.