Arithmetic and Geometric Sequences: Choose the Correct Rule

An arithmetic sequence repeats an addition; a geometric sequence repeats a multiplication. Before choosing a formula, test which operation stays the same. Then decide whether the question asks for one term or the total of several terms. Those two decisions prevent most indexing and formula errors.

These arithmetic and geometric sequences examples use original, constructed numbers for KCSE revision. Assume each stated rule continues for the requested number of terms. A few listed numbers alone cannot prove how an unspecified sequence must continue.

Start with differences and ratios

Use consecutive terms in their given order. For a decreasing arithmetic sequence, the common difference is negative. For a geometric sequence with nonzero terms, divide a later term by the immediately preceding term, not the other way round. Check more than one pair.

A sequence can change sign and still be geometric. For instance, −6, 12, −24, 48 has common ratio −2. A constant nonzero sequence, such as 9, 9, 9, is both arithmetic with difference zero and geometric with ratio one. The familiar categories are not always exclusive.

Keep the term number separate from the term value

Let a be the first term, n the position counted from 1, d the common difference and r the common ratio. Let T_n mean the nth term and S_n mean the sum of the first n terms. The subscript identifies a position or a count; it is not a multiplier.

• Arithmetic term: T_n = a + (n − 1)d.
• Arithmetic sum: S_n = n[2a + (n − 1)d]/2, or n(a + T_n)/2.
• Geometric term: T_n = arⁿ⁻¹.
• Finite geometric sum: S_n = a(1 − rⁿ)/(1 − r), provided r ≠ 1.
• If r = 1, every term is a, so S_n = na.

There are n − 1 steps from the first term to the nth term. That is why both term formulas contain n − 1. However, the geometric sum contains rⁿ, not rⁿ⁻¹. Substituting n = 1 provides a quick test: either sum formula must return a.

Worked example 1: an arithmetic term and its sum

For the arithmetic sequence 11, 17, 23, 29, …, find the 18th term and the sum of the first 18 terms.

1. Identify a = 11 and d = 6.
2. T₁₈ = 11 + (18 − 1) × 6 = 11 + 102 = 113.
3. S₁₈ = 18(11 + 113)/2 = 9 × 124 = 1,116.

The value 113 belongs to one position. The value 1,116 combines all 18 positions. A useful independent check pairs the first and last terms: 11 + 113 = 124. The second and second-last terms also total 124. Eighteen terms make nine such pairs, giving 9 × 124.

Using 11 + 18 × 6 would give 119, which is the 19th term. Count the gaps rather than the number of labels: positions 1 to 18 contain 17 gaps.

Worked example 2: reconstructing an arithmetic rule

An arithmetic sequence has fourth term 22 and ninth term 47. Find its first term and its 12th term.

Write both facts using the same unknowns: a + 3d = 22 and a + 8d = 47. Subtracting the first equation from the second gives 5d = 25, hence d = 5. Substitute into a + 3d = 22: a = 22 − 15 = 7.

Therefore T₁₂ = 7 + 11 × 5 = 62. Check backwards: 7 + 3 × 5 = 22 and 7 + 8 × 5 = 47, so both supplied conditions are satisfied.

The difference between positions 4 and 9 is five steps, not six. Writing out 4 → 5 → 6 → 7 → 8 → 9 makes this visible without having to list every sequence value.

Worked example 3: a shrinking geometric sequence

Find T₇ and S₇ for 96, 48, 24, 12, …, with each term half its predecessor.

Here a = 96 and r = 1/2. T₇ = 96(1/2)⁶ = 96/64 = 1.5. For the sum, S₇ = 96[1 − (1/2)⁷]/[1 − 1/2] = 96 × (127/128) × 2 = 190.5.

Check by direct addition: 96 + 48 + 24 + 12 + 6 + 3 + 1.5 = 190.5. Although the individual terms shrink, their running total grows because all terms are positive. A decreasing sequence does not necessarily have a decreasing sum.

Worked example 4: count the whole arrangement

A fictional hall has 12 rows. The first row has 18 seats, and each following row has three more seats. How many seats are there altogether?

The row sizes form an arithmetic sequence with a = 18, d = 3 and n = 12. The last row contains 18 + 11 × 3 = 51 seats. Total seats = 12(18 + 51)/2 = 414 seats.

This model assumes that no seats are removed for aisles and that all 12 rows follow the stated rule. The average row size is (18 + 51)/2 = 34.5. An average can be fractional even though each actual row contains a whole number of seats. Multiplying that average by 12 returns a whole-number total.

Try three independent questions, then inspect the reasoning

Practice 1: negative first term

Find the 16th term and first-16-term sum of −5, 2, 9, …, assuming an arithmetic rule.

Solution: d = 7. T₁₆ = −5 + 15 × 7 = 100. S₁₆ = 16(−5 + 100)/2 = 8 × 95 = 760. The first negative term must remain in the total; replacing it with positive 5 would change the sequence.

Practice 2: rapid geometric growth

Find the sixth term and the first-six-term sum of 5, 15, 45, …, assuming a geometric rule.

Solution: r = 3. T₆ = 5 × 3⁵ = 1,215. S₆ = 5(3⁶ − 1)/(3 − 1) = 5 × 728/2 = 1,820. The equivalent sum formula with r − 1 in the denominator requires rⁿ − 1 in the numerator, preserving both signs.

Practice 3: two known positions

An arithmetic sequence has T₃ = 14 and T₈ = 39. Find S₁₀.

Solution: 5d = 39 − 14 = 25, so d = 5. Then a = 14 − 2 × 5 = 4. The tenth term is 4 + 9 × 5 = 49. Thus S₁₀ = 10(4 + 49)/2 = 265. Substitute into both known terms before trusting the final sum.

Frequently asked questions

Is a series different from a sequence?

Yes. A sequence lists ordered terms; a series adds them. “The eighth term” and “the sum of eight terms” therefore ask for different quantities.

Can a common difference be negative?

Yes. Subtracting a fixed amount is adding a negative difference. Keep that sign when substituting into the arithmetic formulas.

Does a negative ratio make every term negative?

No. With a nonzero first term, a negative ratio alternates the signs. A positive ratio preserves the sign of the first term.

Why is dividing by the first term sometimes invalid?

Division by zero is undefined. If a displayed term is zero, inspect the stated generating rule instead of applying the consecutive-ratio test blindly.

Can I use an infinite-sum formula here?

These questions ask for finite sums. An infinite geometric sum additionally requires |r| < 1 and an explicitly infinite process; do not substitute it for a requested finite total.

What if I find a non-integer number of terms?

A count of terms must be a positive integer. Recheck the algebra and conditions; the proposed last value may not occur in that sequence.

What should I write before using a formula?

State the pattern, list a and d or r, and identify whether n is a position or a count. Then check at least one known term.

Continue revising

Use the existing Mathematics topical revision resource for the site’s broader subject collection. Its content is login-gated; this lesson does not claim that a particular downloadable test is available.

Reference checks: OpenStax on arithmetic sequences and geometric sequences and series. All worked questions and contextual data above were constructed for this lesson.