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:

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:

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: