Geometric Progression Problems

[MathDefinition title="Geometric Progression (GP)"] A geometric progression is a sequence where each term after the first is obtained by multiplying the previous term by a fixed non-zero constant called the common ratio (r). [/MathDefinition]

[MathDefinition title="nth Term of GP"] The nth term of a GP is given by: a_n = a * r^(n-1) where a is the first term and r is the common ratio. [/MathDefinition]

[MathKeyFormula title="Key Formulas for GP"]

1. nth term: a_n = a * r^(n-1)
2. Common ratio: r = a_{n}/a_{n-1}
3. Sum of n terms (r ≠ 1): S_n = a(r^n - 1)/(r - 1) = a(1 - r^n)/(1 - r)
4. Sum of infinite GP (|r| < 1): S_inf = a/(1 - r)
5. Geometric mean of a and b: √(ab)
6. Three consecutive terms: a/r, a, ar [/MathKeyFormula]

[MathNote title="Important Properties"]

1. The product of terms equidistant from the beginning and end is constant
2. If a, b, c are in GP, then b² = ac
3. For infinite GP to converge, |r| must be less than 1 [/MathNote]

[MathExample title="Example 1: Finding nth Term"] Problem: Find the 8th term of the GP: 2, 6, 18, 54, ...

Solution: Here a = 2, r = 6/2 = 3, n = 8

Using a_n = a * r^(n-1): a_8 = 2 * 3^(8-1) = 2 * 3^7 = 2 * 2187 = 4374

Therefore, the 8th term is 4374. [/MathExample]

[MathExample title="Example 2: Finding Sum of GP"] Problem: Find the sum of the first 10 terms of the GP: 3, 6, 12, 24, ...

Solution: Here a = 3, r = 6/3 = 2, n = 10

Using S_n = a(r^n - 1)/(r - 1): S_10 = 3(2^10 - 1)/(2 - 1) = 3(1024 - 1)/1 = 3 * 1023 = 3069

Therefore, the sum of the first 10 terms is 3069. [/MathExample]

[MathExample title="Example 3: Infinite GP"] Problem: Find the sum of the infinite GP: 1 + 1/2 + 1/4 + 1/8 + ...

Solution: Here a = 1, r = 1/2

Since |r| = 1/2 < 1, the infinite GP converges.

Using S_inf = a/(1 - r): S_inf = 1/(1 - 1/2) = 1/(1/2) = 2

Therefore, the sum of the infinite GP is 2. [/MathExample]

[MathExample title="Example 4: Finding GP from Conditions"] Problem: The 4th term of a GP is 54 and the 7th term is 1458. Find the GP.

Solution: Let the GP be a, ar, ar^2, ar^3, ...

Given: a * r^3 = 54 ... (1) a * r^6 = 1458 ... (2)

Dividing (2) by (1): r^3 = 1458/54 = 27 r = 3

From (1): a * 27 = 54, so a = 2

The GP is: 2, 6, 18, 54, 162, 486, 1458, ... [/MathExample]