Geometric Progression (GP) is a sequence where each term is multiplied by a constant value, known as the common ratio. This section explores formulas for the nth term and the sum of the first n terms of a GP, the defining property of GPs, and examples of finding terms and sums.

**Key Concepts**

**Definition**: A sequence where each term after the first is obtained by multiplying the previous term by a constant value, known as the common ratio. Example: 2, 4, 8, 16, 32...**Nth Term Formula**: $u_n = a \times r^{n-1}$- $u_n$: nth term of the sequence
- $a$: First term of the sequence
- $n$: Number of terms
- $r$: Common Ratio

**Sum of First n Terms**: $S_n = \frac{a(1 - r^n)}{1 - r}$- $S_n$: Sum of the first n terms

**Sum to Infinity**for ( |r| < 1 ): $S_{\infty} = \frac{a}{1 - r}$

Image courtesy of Cuemath

**Examples**

**Example 1: Finding a Specific Term in a Geometric Progression**

**Question: **Calculate the profit for the year 2008 for a company with an initial profit of £250,000 in 2000, increasing annually by 5%.

**Solution:**

Profit for 2008 (9th term) in the GP:

$u_9 = 250000 \times 1.05^{9-1} = £369,363.86 \text{ (approximately)}$**Example 2: Finding the Sum of First n Terms in a Geometric Progression**

**Question:** Determine the total profit from 2000 to 2009 under a 5% annual increase.

**Solution:**

Total profit for 10 years:

$S_{10} = \frac{250000(1 - 1.05^{10})}{1 - 1.05} = £3,144,473.13 \text{ (approximately)}$**Example 3: Comparing Geometric with Arithmetic Progression**

**Question: **For an equal total profit in 10 years under a constant annual increase (Plan B), find the value of $D$.

**Solution: **

Equating the sum under Plan A to that of an arithmetic progression:

$3,144,473.13 = \frac{1}{2} \times 10 \times [2 \times 250000 + (10-1) \times D]$$D = £14,321.63 \text{ (approximately)}$Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.