For the series 1 + 3 + 3² + ... , the sum to n terms is 3280. Find the value of n.

Concept:

Geometric Progression (GP): The series of numbers where the ratio of any two consecutive terms is the same, is called a Geometric Progression.

• A Geometric Progression of n terms with first term a and common ratio r is represented as:

  a, ar, ar², ar³, ..., ar^n-2, ar^n-1.

• The sum of the first n terms of a GP is:

  Sn = \(\frac{a(r^n-1)}{r-1}\) if r > 1  or Sn = \(\frac{a(1 -r^n)}{1-r}\) if r < 1

Calculation:

For the given geometric series 1 + 3 + 32 + ..., we have a = 1 and r = 3.

Let the sum of first n terms be equal to 3280.

∴ Sn = \(\rm 1\left(\frac{3^n\ -\ 1}{3\ -\ 1}\right)\) = 3280

⇒ \(\rm \left(\frac{3^n\ -\ 1}{2}\right)\) = 3280

⇒ 3ⁿ - 1 = 3280 × 2

⇒ 3ⁿ - 1 = 6560

⇒ 3ⁿ = 6561 = 3⁸

⇒ n = 8.