Selected GP terms form an AP

The fifth, tenth and twentieth terms of a convergent geometric progression, G, are the first three consecutive terms of an arithmetic progression, A.

(i) Determine the common ratio of G. marks

Given that the first term of G is 2,

(ii) evaluate the sum to infinity for G marks;

(iii) find the sum of the first 10 odd-numbered terms in A. marks

TJC JC2 2025 H2 Maths MSM; no separate exam attribution printed for Q5

Total marks: 8

Solution

(i)

Equal differences

Find the common ratio rr of the convergent GP whose 5th, 10th and 20th terms are the first three terms of an AP.

Explore this idea

Hint

  1. Use G10−G5=G20−G10G_{10}-G_5=G_{20}-G_{10}.

  2. Factor the equation before choosing the nonzero branch; then set t=r5t=r^5.

Answer · (i)

r=[(5−1)/2]1/5≈0.90824r=[(\sqrt5-1)/2]^{1/5}\approx0.90824; also literal degenerate branch r=0r=0.

Explore the idea

A GP with a = 2 and r ≈ 0.908

Drag the slider to change n and compare the plotted values.

Indexn = 10
Try an example
term uₙ at n = 10
0.8411
Key idea

The terms at n=5,10,20n=5,10,20 decrease by equal steps: they form an AP.

Ideas to try
Read off u5u_5, u10u_{10} and u20u_{20}: their consecutive differences are equal.
(ii)

Sum to infinity

With first term 2, evaluate the sum to infinity of GG.

Hint

  1. S∞=21−rS_\infty=\frac2{1-r}.

Answer · (ii)

Intended nonzero ratio: 2/(1−r)≈21.82/(1-r)\approx21.8; for r=0r=0, sum 22.

(iii)

Sum of terms of an AP

Find the sum of the first 10 odd-numbered terms in AA.

Hint

  1. AA has first term 2r42r^4 and difference 2r9−2r42r^9-2r^4.

  2. The odd-numbered terms have difference double that.

Answer · (iii)

Intended branch: 10(2r4+9[2r9−2r4])≈−33.210(2r^4+9[2r^9-2r^4])\approx-33.2; for r=0r=0, sum 00.

Takeaway

Factor before dividing

Cancelling a factor can discard a valid family. State the branch used for the printed answer and carry excluded cases through the later parts.