Distinguish repeated addition from repeated multiplication

Try

A fictional display adds a row of lights each day. The first row contains 4 lights, and each later row contains 3 more. Find the fifth row’s size and the total through day 5. The row sizes are 4, 7, 10, 13 and 16; their total is 50. A term and a cumulative sum answer different questions.

Predict

Predict what changes if each new row instead contains twice as many lights as the preceding row. The sizes become 4, 8, 16, 32 and 64. A constant difference has become a constant ratio. The first term remains 4, but neither later terms nor totals match the additive design.

Experiment

Track both the current term and cumulative total.

Index n	Arithmetic term	Arithmetic total	Geometric term	Geometric total
1	4	4	4	4
2	7	11	8	12
3	10	21	16	28
4	13	34	32	60
5	16	50	64	124

Explain why the fifth arithmetic term uses four additions, not five. Then write a rule that generates each next row from the previous row. This recursion is useful for stepwise computation; an explicit rule is useful for jumping directly to a distant index.

Understand

With first term a₁ and common difference d, an arithmetic sequence satisfies aₙ=a₁+(n−1)d and aₙ₊₁=aₙ+d. The index n is a positive integer. A formula with a₀ as its starting value would instead use aₙ=a₀+nd. Mixing these conventions produces an off-by-one error.

The sum of the first n arithmetic terms is Sₙ=n(a₁+aₙ)/2. To derive it, write the sum forward and backward. Each of the n aligned pairs adds to a₁+aₙ, so twice the sum is n(a₁+aₙ). This pairing argument also works for odd n; the middle term is counted twice in the doubled sum.

A geometric sequence with ratio r satisfies aₙ=a₁r^(n−1) and aₙ₊₁=raₙ. A negative ratio alternates signs. A zero ratio makes every term after the first zero; specify the first term directly rather than relying on an ambiguous zero-to-zero-power convention.

For the finite sum Sₙ=a+ar+…+ar^(n−1), multiplying by r shifts the terms. Subtracting leaves (1−r)Sₙ=a−arⁿ. When r≠1, Sₙ=a(1−rⁿ)/(1−r). When r=1, every term equals a and the sum is na. The original sum exists in that case even though the divided formula does not apply.

These are finite-sum statements and do not require a small ratio. A finite geometric sum with r=2 is perfectly valid. Whether an infinite series converges is a separate question requiring additional limiting reasoning. Do not apply an infinite-sum formula merely because the sequence is geometric.

Apply

The savings practice deliberately uses exact unrounded model deposits. Real payments require a cent-rounding policy, which can change the final total. State whether rounding occurs each month or only when presenting a total instead of silently choosing one.

For each sequence question, record the starting index, first term, change rule, requested term or sum, and units. To check a formula, compare its first two terms with the recursion, then derive or justify its general form. Agreement at two terms alone does not prove an arbitrary sequence follows that rule.

Continue

This completes the twelve-lesson functions unit, not the full Algebra II or AP Precalculus curriculum. Trigonometric and polar functions, further inequalities, modeling validation and cumulative assessment still require their own work. Reuse these domain, representation and exact-reasoning habits when entering those sequences.