For the following geometric sequence find the explicit formula. {12, -6, 3, ...}

Solution:

The explicit formula for geometric sequence is of the form aₙ = a₁.r^{n - 1} where, a₁ is the first term and ‘r’ is the common ratio.

Here, a = 12, r = (a₂/a₁) = (-6/ 12) = -1/2, where n is the number of terms

∴ aₙ = 12 × (-1/2)^{n - 1}

Example: Find the 7th term and general term of geometric progression 4, -12, 36, -108,...

Given sequence is a geometric progression a = 4, r = (-12/4) = -3 

∴ General term : aₙ = a₁.r^{n - 1} = 4 × (-3)^{n - 1}

7th term : a₇ = 4 × (-3)^{7 - 1} = 4 × (-3)⁶ = 4 × 729 = 2916

For the following geometric sequence find the explicit formula. {12, -6, 3, ...}

Summary:

Explicit formula of given geometric progression aₙ = 12 × (-1/2)^{n - 1}.