Arithmetic and Geometric Sequences

1Summary 重點摘要

Learn arithmetic and geometric sequences, find nth term and sum.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Recognising arithmetic sequencesAn arithmetic sequence is one in which every term after the first differs from the previous term by the same fixed number d, called the common difference. For example, 1, 3, 5, 7, 9 has common difference 2. To test a sequence, subtract consecutive pairs of terms and confirm the differences are all equal. The common difference may be negative: 10, 7, 4, 1 has d = -3 and decreases; when d = 0 every term is the same, and such a sequence is still arithmetic.
2. Step 2: The nth term of an arithmetic sequenceThe nth term of an arithmetic sequence is Tn = a + (n - 1)d, where a is the first term and d the common difference. The formula adds (n - 1) copies of d because moving from the first term to the nth takes n - 1 steps. For a first term of 2 and a common difference of 3, the 10th term is 2 + 9 × 3 = 29. The most common error is writing n instead of (n - 1); before substituting, confirm which term is required and list a and d clearly.
3. Step 3: The sum of an arithmetic sequenceThe sum of the first n terms of an arithmetic sequence is Sn = n2 × [2a + (n - 1)d]; when the first and last terms are known, the alternative form Sn = n2 × (first term + last term) may be used. The two formulas are equivalent, and the choice depends on the data given: use the first when a and d are known, the second when both end terms are known. The formula comes from pairing terms at the two ends, where each pair has the same sum and there are n2 pairs; understanding this origin makes the formula easy to remember and use.
4. Step 4: Recognising geometric sequencesA geometric sequence is one in which every term after the first is obtained by multiplying the previous term by the same fixed number r, called the common ratio. For example, 1, 2, 4, 8, 16 has common ratio 2. Test by dividing each term by the one before it. When all terms share the same sign, r is positive; a negative r makes the signs alternate, as in 1, -2, 4, -8. The common ratio cannot be 0, otherwise every term from the second onwards becomes 0 and the idea of a ratio collapses.
5. Step 5: The nth term of a geometric sequenceThe nth term of a geometric sequence is Tn = a × r^(n-1), where a is the first term and r the common ratio. As with the arithmetic formula, the exponent is n - 1 rather than n, because the first term has not yet been multiplied by any ratio. For a first term of 3 and a ratio of 2, the 5th term is 3 × 2^4 = 48. Watch the order of operations: evaluate r^(n-1) first and then multiply by a, and for large exponents write the steps separately to avoid errors.
6. Step 6: The sum of a geometric sequenceThe sum of the first n terms of a geometric sequence is Sn = a × (r^n - 1) / (r - 1), valid when r is not 1; when r = 1 every term equals a and the sum is simply Sn = na. Substitute r and n carefully, and always bracket fractional or negative values of r. The formula arises from the shift-and-subtract trick: subtracting r·Sn from Sn cancels most middle terms, and simplifying what remains produces the formula; knowing this origin helps you apply it correctly.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Arithmetic: each term add fixed d
• Geometric: each term multiply fixed r
• Arithmetic nth term: a + (n-1)d
• Arithmetic sum: n2 × (first + last)
• Geometric nth term: a × r^(n-1)