The term free of x, ratios of terms, and the greatest term
Week 3 · The term free of x, ratios of consecutive terms and of their coefficients, three consecutive coefficients and the greatest term · 3 lessons
2
The ratio between two consecutive terms and between their coefficients
Pure Mathematics Sec 3 Term 1, Unit 1, Lessons 2–3
GoalFind the ratio between two consecutive terms, and between their coefficients.
New words
ratio of consecutive terms · T(r+1)/T(r) = ((n − r + 1)/r) · (a/x) for (x + a)ⁿratio of coefficients · the same ratio with the letters removed
Explanation
Divide a term of (x + a)ⁿ by the one before it. T(r+1) = nCr xⁿ⁻ʳ aʳ and T(r) = nC(r−1) xⁿ⁻ʳ⁺¹ aʳ⁻¹, so T(r+1)/T(r) = [nCr / nC(r−1)] · (a/x). Writing both nCr out with factorials, nCr / nC(r−1) = (n − r + 1)/r. This gives the rule T(r+1)/T(r) = ((n − r + 1)/r) · (a/x). The table checks it for (1 + x)⁶ at x = 1, where the ratios are the quotients of neighbouring numbers of the row 1, 6, 15, 20, 15, 6, 1. For the coefficients of (px + qy)ⁿ, keep only the numbers: the ratio of the coefficient of T(r+1) to that of T(r) is ((n − r + 1)/r) · (q/p). Always say which r you use: for T4 ÷ T3 the top term is T(r+1) = T4, so r = 3. These ratios are the tool for the equations of the next lesson: a given ratio becomes an equation for n, x or an unknown number.
(1 + x)⁶ with x = 1: the formula (6 − r + 1)/r
r
T(r+1) ÷ T(r)
the two numbers
1
6
6 ÷ 1
2
5/2
15 ÷ 6
3
4/3
20 ÷ 15
4
3/4
15 ÷ 20
5
2/5
6 ÷ 15
6
1/6
1 ÷ 6
Worked examples
In (x + 3y)¹⁰ find T4 ÷ T3. Here T4 = T(r+1) with r = 3: ((10 − 3 + 1)/3) · (3y/x) = (8/3)(3y/x) = 8y/x.
In (3x + 2)⁸ find the ratio of the coefficient of T4 to that of T3. With r = 3: ((8 − 3 + 1)/3) · (2/3) = 2 × 2/3 = 4/3. Direct check: 8C3 · 3⁵ · 2³ = 108 864 and 8C2 · 3⁶ · 2² = 81 648, and 108 864 ÷ 81 648 = 4/3.
For which x are T2 and T3 of (2 + x)⁶ equal? We use T3 ÷ T2, which has r = 2: ((6 − 2 + 1)/2) · (x/2) = 5x/4. Setting 5x/4 = 1 gives x = 4/5. Check: T2 = 6 · 2⁵ x = 192x and T3 = 15 · 2⁴ x² = 240x², and 240x² = 192x gives x = 4/5.
Class activity
Test the rule. In (1 + x)⁸ at x = 1 write the coefficients 1, 8, 28, 56, 70, … and check three neighbouring quotients against (8 − r + 1)/r.
