Geometric series sum
A geometric series is a sum where each term is obtained by multiplying the previous term by the same constant ratio. To simplify such a sum, first check that the terms really follow a constant multiplier.
Method
- Identify the first term and the common ratio.
- Count the number of terms in the sum.
- Use the geometric sum formula when appropriate: for first term (a), ratio (r), and (n) terms,
[
S_n = a\frac{1-r^n}{1-r} \quad (r\ne 1).
]
- Simplify carefully, keeping signs and powers accurate.
Special case
- If the ratio is (1), every term is the same, so the sum is just the number of terms times that repeated value.
Check your answer
- Make sure the result is simplified.
- If possible, test by adding a few terms directly to see whether the formula gives the same value.
- For negative or fractional ratios, watch the alternating signs and reduce fractions fully.
A good final answer should be exact and clearly simplified, not left in a partially expanded form.