Find the nth Term of a Geometric Sequence

Count how many times the ratio is applied

A geometric sequence multiplies by the same ratio r at each step. Its nth term is an = a1 r^(n − 1). The first term uses exponent zero, so it remains a1. The second has one factor of r, the third has two, and so forth. Multiplying the first term by r^n instead would return the following term. This exponent offset follows from the starting index and should be checked before using a calculator.

Calculate an increasing sequence

For a1 = 3 and r = 2, the sequence begins 3, 6, 12, 24. The eighth term is 3×2^7 = 384. The sum of its first eight terms is 3(1 − 2^8)/(1 − 2) = 765, a distinct result. The local form displays both quantities, so match the requested output. The ratio 2 means a doubling at every step; it is not a common difference of 2.

Handle fractional and negative ratios

With a1 = 80 and r = 1/2, the fifth term is 80(1/2)^4 = 5. With a1 = 2 and r = −3, the fifth term is 2(−3)^4 = 162, while the fourth is −54. Parentheses around a negative base matter when writing the power. Alternating signs are a valid geometric pattern, and a ratio with magnitude below one makes nonzero terms shrink in magnitude rather than adding a fixed decrement.

Recover a ratio only when the information supports it

From a2 = 6 and a5 = 162, r^3 = 162/6 = 27, giving r = 3 and a1 = 2. If the index gap were even, a positive term ratio could permit both positive and negative real ratios until additional terms resolve the sign. Dividing by a known zero term is not valid. State the permitted ratio domain and use all supplied terms to check any candidate instead of selecting a convenient root without justification. Testing another known position can distinguish ratios that otherwise fit only the separated endpoints.

State the first index and the starting value

A sequence assigns a value to each allowed position. In the arithmetic and geometric formulas here, the first position is n = 1 and its value is a1. The local sequence forms follow that convention. If a problem instead begins at index zero, its initial value is a0 and the exponent or increment count shifts accordingly. Write the index beside the starting value before substituting numbers. An otherwise correct formula can be one term out if the starting convention changes between a worksheet, a program and the calculator.

Distinguish a term from a sum

The nth term is the value at one position, while the sum of the first n terms adds all values from the starting position through n. These can be very different numbers. Read the output label before copying an answer. The local sequence tools show both a term and a finite sum, plus a preview of initial terms. A short preview does not mean the calculation stopped at the last displayed preview value. Use a small example that can also be added or generated by hand when checking which output the question requests.

Check that the proposed pattern is justified

A short list can fit more than one mathematical rule. Constant consecutive differences support an arithmetic model, while constant ratios between nonzero consecutive terms support a geometric model. A problem should state the intended pattern or provide enough context to choose it. Do not force an arbitrary measurement series into one of these models merely because a calculator accepts its first two numbers. Preserve fractions or adequate intermediate precision when deriving a difference or ratio, then round only the final requested result. Negative terms and decreasing patterns can be valid; the position n still needs to be a whole number.

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