Geometric Progression and Infinite Sum
A geometric progression has constant ratio r, nth term ar^(n-1), finite sum a(1-rn)/(1-r) and infinite sum a/(1-r) when |r| < 1.
What is Geometric Progression and Infinite Sum?
A geometric progression has constant ratio r, nth term ar^(n-1), finite sum a(1-rn)/(1-r) and infinite sum a/(1-r) when |r| < 1.
Key formula / rule: The geometric mean of two positive numbers is the square root of their product.
Key points
- Compute finite and infinite sums of geometric progressions.
- Evaluate whether an infinite geometric series converges.
Common exam trap
Using the infinite sum formula when |r| ≥ 1.
Definitions
- Term
Geometric Progression and Infinite Sum
- Meaning
A geometric progression has constant ratio r, nth term ar^(n-1), finite sum a(1-rn)/(1-r) and infinite sum a/(1-r) when |r| < 1.
- Term
Geometric Progression and Infinite Sum — explanation
- Meaning
Multiplicative growth appears wherever a quantity scales by a fixed factor each step. The infinite sum converges only for |r| < 1, and that condition is examined as often as the formula itself.
Learning objectives
Compute finite and infinite sums of geometric progressions.
Evaluate whether an infinite geometric series converges.
Formulae
- Key point
The geometric mean of two positive numbers is the square root of their product.
- Key point
The infinite sum exists only when |r| < 1.
- Key point
Three numbers in GP can be written a/r, a, ar.
Prerequisites
MAT-U1-SS-T1-S1-C1
Common mistakes
Using the infinite sum formula when |r| ≥ 1.
Confusing the number of terms with the exponent of r.
Keywords
Geometric
Progression
Infinite
Practice preview
The sum of an infinite GP is 3 and the sum of the cubes of its terms is 27/19. Find the common ratio.…
hard
Find the common ratio of the Geometric Progression 3, 6, 12, 24, ...…
easy
What is the 5th term of a Geometric Progression whose first term is 2 and common ratio is 3?…
easy