Selected GP terms form an AP · source reference

Printed question

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. [3]

Given that the first term of G is 2,

(ii) evaluate the sum to infinity for G [2];

(iii) find the sum of the first 10 odd-numbered terms in A. [3]

Printed working

(i) Let $a$ be the first term of G.

$$ar^9-ar^4=ar^{19}-ar^9\Rightarrow ar^4(r^5-1)=ar^9(r^{10}-1),$$ $$\frac{r^5-1}{(r^5)^2-1}=r^5\Rightarrow\frac{r^5-1}{(r^5-1)(r^5+1)}=r^5\Rightarrow\frac1{r^5+1}=r^5\Rightarrow r^{10}+r^5-1=0.$$

From GC: $r=-1.10$ (rejected since convergence requires $|r|<1$) or $r=0.90824$ (5 s.f.), hence 0.908 (3 s.f.).

(ii) $S_{\infty}=\frac2{1-0.90824}=21.8$ (3 s.f.).

(iii) AP first term $=2r^4$, common difference $=2r^9-2r^4$. The odd-term sum is

$$\frac{10}2[2(2r^4)+9\cdot2(2r^9-2r^4)]=-33.2.$$

Teacher notes

For GP G, $G_n=ar^{n-1}$. AP terms $A_1,A_2,A_3$ are $G_5,G_{10},G_{20}=ar^4,ar^9,ar^{19}$. Equal differences: $A_2-A_1=A_3-A_2$.

AP odd-indexed terms $A_1,A_3,A_5,\ldots$ have difference $2d$. “Odd-numbered” refers to indices: their values need not be odd.

GP infinite sum $a/(1-r)$.

Printed answer key

(i) $r=0.90824$ (5 s.f.) =0.908 (3 s.f.); (ii) 21.8 (3 s.f.); (iii) −33.2

Mathematical review and qualifications

For first term $b$, equal differences give $br^4(r^{15}-2r^5+1)=0$. In the intended nondegenerate case $b \ne0,r \ne0$, put $t=r^5$: $t^3-2t+1=(t-1)(t^2+t-1)=0$. Convergence excludes $r=1$ and the negative root with modulus exceeding one. Thus $r=[(\sqrt5-1)/2]^{1/5}\approx0.90824$.

(ii) With $b=2$, $S_\infty=2/(1-r)\approx21.8$ (3 s.f.).

(iii) $A_1=2r^4$, $d=2r^9-2r^4$. The ten odd-indexed terms have difference $2d$, sum $10[A_1+9d]\approx-33.2$.

Qualification: the literal wording also admits $r=0$, yielding $G=(2,0,0,\ldots)$ and the zero AP. Then (ii) is 2 and (iii) is 0. The printed solution chooses the nonzero branch; that assumption is recorded rather than silently imposed.

The printed solution divides by r^4 and thereby omits the zero-ratio GP. The displayed main answer follows its intended nondegenerate branch.

Provenance

2025 TJC H2 Math (9758) JC2 MSM: ARITHMETIC & GEOMETRIC PROGRESSIONS. TJC JC2 2025 H2 Maths MSM; no separate exam attribution printed for Q5 Question PDF pages 1; solution PDF pages 6, 7. The original files remain private QA records outside this native content.