Geometric Sequence Worksheet: Recursive Rules and Answers

Write the initial value with the recurrence

A recursive geometric rule states a1 and then an = r a(n−1) for n at least 2. The multiplier alone does not determine a unique sequence: doubling can produce 1,2,4 or 5,10,20 depending on the start. A worksheet answer should therefore include both the starting term and the recurrence. The explicit equivalent is an = a1 r^(n − 1), which lets you reach a position without calculating every preceding term.

Exercise one: complete a positive-ratio sequence

Given a1 = 5 and an = 3a(n−1), find the next three terms and a6. Repeated multiplication gives 5, 15, 45 and 135. The sixth term is 5×3^5 = 1,215. Enter first term 5, ratio 3 and position 6 in the local form. The fifth exponent counts five transitions from the first term, while the position remains six. This distinction is a common source of a result that is one multiplication too large.

Exercise two: write a rule for alternating signs

For 4, −8, 16, −32, divide each term by its predecessor to obtain ratio −2. A recursive answer is a1 = 4 and an = −2a(n−1). Its explicit form is 4(−2)^(n − 1). The seventh term is 4×64 = 256. Check that odd positions are positive and even positions negative in this example. An answer using ratio 2 matches magnitudes but fails the signs and therefore does not describe the supplied sequence.

Exercise three: distinguish halving from subtraction

For a1 = 64 and ratio 1/2, the first six terms are 64, 32, 16, 8, 4 and 2. The differences are not constant, so an arithmetic rule such as subtracting 32 would fail after the second term. The sum of these six geometric terms is 126. As a check, pair that finite total with the next omitted term: the remaining infinite tail would total 2 in this particular convergent example, making the full infinite sum 128.

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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