Identify a Geometric Sequence and Find Its Sum

Check ratios between consecutive nonzero terms

For 6, 18, 54 and 162, each later term divided by its predecessor gives 3. The list fits a geometric rule with first term 6 and ratio 3. By contrast, 6, 12, 18 has ratios 2 and 1.5 and is not geometric. The first two numbers alone are not enough to verify the rest of a supplied list. Zero terms need special care because division by zero cannot establish a ratio.

Generate and calculate a term

With a1 = 6 and r = 3, the fifth term is 6×3^4 = 486. Enter those values and n = 5 in the calculator. The first five terms are 6, 18, 54, 162 and 486. Adding them gives 726. This direct list check is useful for a modest number of terms and distinguishes the final term 486 from the sum 726. For very long sequences, the explicit formulas avoid repeated manual multiplication.

Use the finite-sum formula

For r different from 1, S_n = a1(1 − r^n)/(1 − r). The example gives 6(1 − 243)/(1 − 3) = 726. If r = 1, every term equals a1 and the sum is na1; substituting into the fraction would produce a zero denominator even though the sum is perfectly well defined. A zero first term gives an all-zero sequence under a finite multiplication rule. Keep such special cases separate from an attempt to infer ratios by dividing zero terms.

Recognize when an infinite sum exists

For a nonzero geometric sequence with |r| < 1, the infinite sum is a1/(1 − r). Starting at 10 with ratio 0.2 gives an infinite sum of 12.5, while the sum of the first three terms 10 + 2 + 0.4 is 12.4. The remaining tail is 0.1. A finite sum must not be labelled infinite merely because later terms are small. Ratios with magnitude at least one do not give this convergent infinite sum for a nonzero start.

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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