Summary — What This Chapter Covers
- (a + b)n for positive integer n, using nCr = n!/(r!(n − r)!).
- AP: un = a + (n − 1)d and Sn = ½n[2a + (n − 1)d].
- GP: un = arn−1 and Sn = a(1 − rn)/(1 − r).
- A GP converges if |r| < 1, and then S∞ = a/(1 − r).
- a, b, c are in AP if 2b = a + c, and in GP if b2 = ac.
1. The Binomial Expansion
Binomial
(a + b)n for positive integers n
The term in br is nCran−rbr. Put any coefficient of x inside the bracket with x when you raise it to a power: (3x)3 = 27x3, not 3x3. For "term independent of x", write the general term and make the power of x zero.
Binomial theorem
(a + b)n = an + nC1an−1b + nC2an−2b2 + … + bn, nCr = n!/(r!(n − r)!)
Worked Example
Find the coefficient of x3 in the expansion of (2 + 3x)5.
Practice Question 1
Find the first three terms, in ascending powers of x, of (1 − 2x)6.
Practice Question 2
Find the term independent of x in (x + 2/x)6.
2. Arithmetic Progressions
AP
Constant difference d
Each term is the previous one plus d. Two pieces of information (two terms, or a term and a sum) give two equations in a and d. "How many terms" questions lead to an inequality in n; n must be a whole number.
AP formulae
un = a + (n − 1)d · Sn = ½n[2a + (n − 1)d] = ½n(a + l), l = last term
Worked Example
The 5th term of an AP is 18 and the sum of the first 10 terms is 210. Find a and d.
Practice Question
Find the least number of terms of 3 + 7 + 11 + … needed for the sum to exceed 500.
3. Geometric Progressions and Sum to Infinity
GP
Constant ratio r
Each term is the previous one multiplied by r. If |r| < 1, the terms shrink towards zero and the sum approaches a/(1 − r); if |r| ≥ 1, there is no sum to infinity. Questions often combine an AP and a GP: write each condition as an equation and solve.
GP formulae
un = arn−1 · Sn = a(1 − rn)/(1 − r) · S∞ = a/(1 − r), |r| < 1