Eliminate two progression parameters · source reference

Printed question

An arithmetic series has first term $a$ and common difference $d$, where $a,d$ are non-zero. A convergent geometric series has first term $b$ and common ratio $r$, where $b$ is positive and $r$ is non-zero. The fourth and ninth terms of the arithmetic series equal the sixth and ninth terms of the geometric series respectively, and the eleventh term of the arithmetic series is $br^6$ less than the fourteenth term of the geometric series.

(i) Show $r$ satisfies $5r^8-7r^3-5r+2=0$ and solve, giving your answer correct to 4 decimal places. [4]

(ii) Using this value of $r$, deduce that for any positive integer $n$, the sum of the terms of the geometric series after, but not including, term $n$ is less than $\frac35b$. [2]

Printed working

(i) From AP $u_n=a+(n-1)d$ and GP $u_n=br^{n-1}$,

$$a+3d=br^5\quad(1),\qquad a+8d=br^8\quad(2),\qquad a+10d=br^{13}-br^6\quad(3).$$

$(2)-(1)$: $d=\frac b5(r^8-r^5)$. $(3)-(2)$: $d=\frac b2(r^{13}-r^8-r^6)$.

$$\frac b5(r^8-r^5)=\frac b2(r^{13}-r^8-r^6).$$

Since $b\ne0,r\ne0$,

$$2r^8-2r^5=5r^{13}-5r^8-5r^6\Rightarrow2r^3-2=5r^8-5r^3-5r\Rightarrow5r^8-7r^3-5r+2=0.$$

GC real roots: 1.1371 and 0.34347. Since convergent, $r=0.3435$ (4 d.p.).

(ii) Required sum $=\frac{u_{n+1}}{1-r}=\frac{b(0.3435)^{n+1-1}}{1-0.3435}\approx1.523b(0.3435)^n\le1.523b(0.3435)^1\approx0.52315b<\frac35b$.

Alternatively required sum $=S_{\infty}-S_n=\frac b{1-r}-\frac{b(1-r^n)}{1-r}=\frac{br^n}{1-r}$.

Printed answer key

(i) $r=0.3435$ (4 d.p.)

Mathematical review and qualifications

(i) Subtracting yields $5d=b(r^8-r^5)$ and $2d=b(r^{13}-r^6-r^8)$. Equating $10d$ and cancelling $br^5 \ne0$ gives $2(r^3-1)=5(r^8-r-r^3)$, hence $5r^8-7r^3-5r+2=0$. Its real roots are approximately $0.34347,1.1371$; convergence rejects the latter, so $r=0.3435$ (4 d.p.).

(ii) Retain the unrounded root. The tail is $br^n/(1-r)\le br/(1-r)\approx0.5232b<0.6b=3b/5$.

Provenance

2025 TJC H2 Math (9758) JC2 MSM: ARITHMETIC & GEOMETRIC PROGRESSIONS. [2024/NJC/Prelim/II/2] Question PDF pages 2; solution PDF pages 9, 10. The original files remain private QA records outside this native content.