A relative geometric-tail error

The difference between the sum of the first nn terms of the geometric progression and its sum to infinity is less than 0.4% of the sum to infinity. Assuming the value of rr is 34\frac34, find the least value of nn. marks

[2022/RVHS/Promo/3(ii)]

Solution

Least number of terms

Geometric tail

The difference between SnS_n and S∞S_\infty is less than 0.4% of S∞S_\infty and r=34r=\frac34. Find the least nn.

Explore this idea

Hint

  1. Write S∞−Sn=S∞rnS_\infty-S_n=S_\infty r^n.

  2. Solve (34)n<0.004\left(\frac34\right)^n<0.004 with logarithms.

Answer · Least number of terms

2020 terms.

Percentage error assumes a nonzero positive total.

Explore the idea

The tail fraction (3/4)ⁿ against the 0.4% line

Compare the fraction still missing with 0.004. Focus on the adjacent counts 19 and 20.

Indexn = 20
Try an example
tail fraction rⁿ at n = 20
0.003171
Key idea

The tail fraction first drops below 0.004 at n=20n=20.

Ideas to try
Compare n=19n=19 and n=20n=20 using the readout. Which is the first count below 0.004?
Takeaway

Check both sides of an integer boundary

A logarithm gives a real threshold. Substitute the neighbouring integer counts into the original strict inequality to establish the least count.