Geometric series 3, 6, 12: sum of 7 terms

The first term of a geometric sequence is 3, the common ratio is 2, find the sum of its first 7 terms.

Step-by-step

1. First term a_1 = 3, common ratio q = 2
   In a geometric sequence, the ratio of adjacent terms is constant; this constant is the common ratio.
2. aₙ = a₁ · q^(n−1) = 3 × 2^6 = 192
   General term: starting from the first term, multiply by the common ratio n − 1 times
3. q^6 = 64
   Compute the power of the common ratio first, then multiply by the first term to avoid multiplication errors.
4. Sₙ = a₁(qⁿ − 1)/(q − 1) = 381
   The geometric sum formula works only when the common ratio is not 1

Why it works

The formula for the sum of a geometric sequence comes from the method of subtracting shifted terms: multiply the sum by the common ratio and subtract it from the original expression, so that intermediate terms cancel in pairs. The denominator is the common ratio minus one; when the common ratio is 1 or 0, the formula fails and must be checked first. After calculating the result, you can manually add the first few terms to verify the order of magnitude, since geometric sequences grow quickly and the last term is often much larger than the first.

Common mistakes

• Confusing the geometric sequence sum formula with the arithmetic sequence sum formula.
• Writing the numerator as the common ratio raised to the number of terms minus one, but forgetting to multiply by the first term.
• Still forcing the formula when the common ratio is 1 or 0, causing division by zero.