Finite GP with unknown index

This problem focuses on finite geometric series, a cornerstone of the Sequences and Series topic in HSC Mathematics Extension 1. By working through it, you’ll practise applying the sum formula for a GP and solving for the number of terms when the sum is known — a skill that blends quick algebraic manipulation with recognising powers of the common ratio.

Problem Statement

For the geometric sequence with first term $5$ and ratio $2$, find $n$ such that

$$S_n=2555.$$

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Solutions

We begin by writing the formula for the sum of a finite geometric series. With first term $a = 5$ and common ratio $r = 2$, we substitute directly:

$$S_n=5\frac{2^n-1}{2-1}=5(2^n-1)=2555.$$

The equation simplifies nicely; the job now is to isolate $2^n$. Now we solve for $n$. Setting the expression equal to 2555, divide both sides by $5$ to isolate the power term:

$$2^n-1=511 \implies 2^n=512=2^9.$$

Since the base is the same on both sides, we can equate the exponents to obtain $n=9$.

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Takeaways

• The finite sum formula $S_n = a\frac{r^n-1}{r-1}$ becomes especially neat when $r = 2$, because the denominator simplifies to $1$.
• Solving for $n$ involves isolating the exponential term and then recognising it as a power of the common ratio — no logarithms are needed when the numbers work out this cleanly.
• Always check that your final $n$ is a positive integer, since it represents the number of terms in the sequence.

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Further Readings

HSC Distributions, HSC Integrals, HSC Probability, HSC Inequalities