Zadanie 1g
The two bounds to verify are and . Again un = (2n + 1)3 − (2n − 1)3. (i) Express u_n as a quadratic in v = 2n + 1. (ii) Hence express in terms of T2(n), T1(n), T0(n). (iii) Hence show that…
Finite binomial expansions
Expand a binomial with a positive integer power and determine individual coefficients or terms.
Tę umiejętność sprawdza 15 zadań wzorowanych na arkuszach IB z lat 2022–2026.
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Zadanie 2b
Expand (1 + t)4 and collect like powers of t.
Zadanie 2c
Hence or otherwise, find the balance after one year, rounded to the nearest dollar.
Zadanie 11d
Expand (1 + z)6 + (1 − z)6 and use its roots to prove .
Zadanie 2
Find the coefficient of x5 in (3x − 2)8.
Zadanie 4
For , write (x + h)7 = x7 + ax6 + bx5 + cx4 + dx3 + ⋯ + h7. The coefficients a, b, d, in that order, form a geometric sequence. Find h.
Zadanie 5
For a,b>0, the coefficient of x⁴ in (ax2 + b)7 is 6048, and its coefficient in (ax2 + b)9 is 41472. Find a and b.
Zadanie 5
The expansion of (1 + kx)n begins 1 + 15x + 45k2x2 + ⋯ + knxn, with positive integer n and rational k. Find n and k.
Zadanie 5
The expansion of (1 + kx)n begins 1 + 15x + 45k2x2 + ⋯ + knxn, with positive integer n and rational k. Find n and k.
Zadanie 12a
Expand in the form A+iB, with A and B expressed using c,s.
Zadanie 12a
Expand in the form A+iB, with A and B expressed using c,s.
Zadanie 5a
Show that b=40.
Zadanie 5b
The third term is the arithmetic mean of the second and fourth terms. Find all possible x.
Zadanie 6
For a positive integer n, consider (5x2 − 2/x3)n. Determine every n for which the expansion contains a nonzero constant term.
Zadanie 5
For a≠0, the coefficient of x3 in (1 + ax)8/(14x2) is 12a4. Find a.