Eliminate two progression parameters

An arithmetic series has first term aa and common difference dd, where a,da,d are non-zero. A convergent geometric series has first term bb and common ratio rr, where bb is positive and rr 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 br6br^6 less than the fourteenth term of the geometric series.

(i) Show rr satisfies 5r8−7r3−5r+2=05r^8-7r^3-5r+2=0 and solve, giving your answer correct to 4 decimal places. marks

(ii) Using this value of rr, deduce that for any positive integer nn, the sum of the terms of the geometric series after, but not including, term nn is less than 35b\frac35b. marks

[2024/NJC/Prelim/II/2]

Total marks: 6

Solution

(i)

Eliminating parameters

Show 5r8−7r3−5r+2=05r^8-7r^3-5r+2=0 and solve it, giving rr to 4 d.p.

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Hint

  1. a+3d=br5a+3d=br^5, a+8d=br8a+8d=br^8, a+10d=br13−br6a+10d=br^{13}-br^6.

  2. Subtract to find dd two ways.

Answer · (i)

5r8−7r3−5r+2=05r^8-7r^3-5r+2=0; convergent root r=0.3435r=0.3435 (4 d.p.).

Explore the idea

Keep the root inside the convergence interval

Reveal the roots, then compare them with −1 < r < 1. This is a continuous polynomial graph; the horizontal coordinate is r.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

Ideas to try
Switch on both the roots and the convergence interval. Which root satisfies both conditions?
(ii)

Bounding the tail

Deduce that for every positive integer nn the sum of the terms after term nn is less than 35b\frac35b.

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Hint

  1. Tail =brn1−r=\frac{br^n}{1-r}.

  2. Its largest value is at n=1n=1.

Answer · (ii)

Tail brn/(1−r)≤br/(1−r)≈0.5232b<3b/5br^n/(1-r)\le br/(1-r)\approx0.5232b<3b/5.

n∈Z+n\in\mathbb Z^+; use the unrounded root.

Explore the idea

The tail after n terms (b = 1)

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

Indexn = 1
Try an example
tail after n at n = 1
0.5232
Key idea

Every tail is below 35b\frac35b: the largest is 0.523b0.523b.

Ideas to try
Is the largest tail at n=1n=1?
Takeaway

A bound for every n needs a maximum

Checking a few tails is not enough. Prove the tail decreases, identify its largest allowed value, then compare that value with the required bound.