Translating Fixed Increments and Percentage Growth

Why the Type of Change Matters

A payment schedule may be described as “increasing by $60 each year” or “increasing by 6% each year.” These instructions sound similar, but they produce different payment sequences. A fixed dollar increase adds the same amount each period; a fixed percentage increase applies the same proportion to the previous payment. Identifying which rule governs the payments is the first step in representing a varying annuity correctly.

This tutorial focuses on translating the wording or observed amounts into a payment sequence. Assume payments occur at regular intervals and are known with certainty. Unless an example says otherwise, no interest rate, mortality assumption, or probability distribution is involved: the task is to classify and write the payment amounts, not to value them.

Definition: Let $C_k$ be the amount of the $k$th payment, where $k=1,2,\ldots,n$. A payment sequence is arithmetic if each payment differs from the preceding payment by the same dollar amount $G$: $$ C_k=C_1+(k-1)G. $$ It is geometric if each payment is the preceding payment multiplied by the same factor $1+r$, where $r$ is the percentage change per period: $$ C_k=C_1(1+r)^{k-1}. $$

The first payment is $C_1$ in both expressions. The exponent or multiplier counts changes after that first payment: the second payment has undergone one change, and the $k$th payment has undergone $k-1$ changes. A positive $G$ or $r$ means the payments increase; a negative value means they decrease. If $G=0$ or $r=0$, the payments are level, and the level sequence fits both descriptions.

Test Differences and Ratios

When amounts are listed, compare adjacent payments. For an arithmetic sequence, the differences $C_{k+1}-C_k$ must be constant. For a geometric sequence, the ratios $C_{k+1}/C_k$ must be constant, provided the denominator is nonzero. A constant ratio of $1.04$, for example, means each payment is 4% greater than the one immediately before it.

The two tests ask different questions. Differences are measured in dollars; ratios are unitless and measure proportional change. It is not enough to notice that the payments rise. Nor does one matching difference or ratio establish a pattern over several periods: check all available consecutive pairs.

Locate the first payment.
Identify its amount and the payment number to which it belongs. Do not accidentally treat the second payment as the base.

Translate the change rule.
“Add $G” specifies a fixed dollar change. “Increase by $r$% of the preceding payment” specifies a fixed percentage change.

Write and test the sequence.
Use the arithmetic or geometric expression and verify it against the stated amounts or wording.

Why the Sequence Formulas Hold

The formulas follow by applying the same one-period rule repeatedly. In an arithmetic pattern, every step adds $G$. Thus the first step gives $C_2=C_1+G$, the second gives $C_3=C_1+2G$, and after $k-1$ steps,

C_k=C_1+\underbrace{G+\cdots+G}_{k-1\text{ additions}}=C_1+(k-1)G.

In a geometric pattern, each step multiplies by $1+r$. Starting with $C_2=C_1(1+r)$, applying the same multiplication $k-1$ times gives

C_k=\underbrace{C_1(1+r)(1+r)\cdots(1+r)}_{k-1\text{ multiplications}} =C_1(1+r)^{k-1}.

This also explains why a percentage increase usually creates dollar increases that get larger over time: the percentage is applied to a changing payment amount.

Worked Examples

Worked Example: A Fixed Dollar Increase

A hypothetical maintenance contract requires five annual payments. The first payment, due at the end of year 1, is $1,350. Each later payment is $85 more than the payment immediately before it. Write the payment sequence and find the fifth payment.

Assumptions and method. Payments are deterministic and annual. No interest rate, mortality assumption, or probability distribution is used. The phrase “$85 more than the payment immediately before it” specifies an arithmetic pattern with $C_1=1{,}350$ and $G=85$.

The general expression is $$ C_k=1{,}350+(k-1)(85). $$ For the five payments, evaluate it at each payment number:

\begin{aligned} C_1&=1{,}350+0(85)=1{,}350,\\ C_2&=1{,}350+1(85)=1{,}435,\\ C_3&=1{,}350+2(85)=1{,}520,\\ C_4&=1{,}350+3(85)=1{,}605,\\ C_5&=1{,}350+4(85)=1{,}690. \end{aligned}

The sequence is $1,350, $1,435, $1,520, $1,605, $1,690. The fifth payment is $1,690. Check: each adjacent difference is $85, including $1{,}690-1{,}605=85$. The method is to apply the fixed-dollar rule from the first payment; it avoids multiplying the first payment by a percentage merely because the amounts increase over time.

Worked Example: A Fixed Percentage of the Previous Payment

A hypothetical equipment lease has four annual payments. The first is $1,000, and each subsequent payment is 10% greater than the immediately preceding payment. Write the sequence and determine the fourth payment.

Assumptions and method. Payments occur annually and are deterministic; no interest rate, mortality assumption, or probability distribution is used. The growth rate is $r=0.10$, applied to the payment in the prior year. Therefore the geometric model is $C_k=1{,}000(1.10)^{k-1}$.

Calculate each payment from the prior one:

\begin{aligned} C_1&=1{,}000,\\ C_2&=1{,}000(1.10)=1{,}100,\\ C_3&=1{,}100(1.10)=1{,}210,\\ C_4&=1{,}210(1.10)=1{,}331. \end{aligned}

The sequence is $1,000, $1,100, $1,210, $1,331, and the fourth payment is $1,331. Check: the successive ratios are $1{,}100/1{,}000=1.10$, $1{,}210/1{,}100=1.10$, and $1{,}331/1{,}210=1.10$. The dollar increases are $100, $110, and $121, so they are not constant. This method avoids incorrectly treating a repeated percentage as a repeated dollar increase.

Worked Example: Classifying a Listed Schedule

A hypothetical service agreement lists annual payments of $800, $840, $882, and $926.10 for years 1 through 4. The agreement also states that each payment after the first is 5% greater than the preceding payment. Confirm the pattern and find the year 5 payment.

Assumptions and method. The listed annual amounts are deterministic; no interest rate, mortality assumption, or probability distribution is involved. Test both differences and ratios, then extend the geometric rule.

The consecutive dollar increases are $$ 840-800=40,\qquad 882-840=42,\qquad 926.10-882=44.10. $$ These are not constant, so the schedule is not arithmetic. The consecutive ratios are $$ \frac{840}{800}=1.05,\qquad \frac{882}{840}=1.05,\qquad \frac{926.10}{882}=1.05. $$ Thus it is geometric with $C_1=800$ and $r=0.05$. Its expression is $C_k=800(1.05)^{k-1}$, and the fifth payment is $$ C_5=800(1.05)^4=800(1.21550625)=972.405. $$ The exact calculated amount is $972.405; to the nearest cent, the year 5 payment is $972.41. Check: the unrounded amount equals $926.10(1.05)=972.405$. The method confirms the rule using every adjacent pair; it avoids concluding that the first $40 increase remains fixed just because it is the first observed increase.

Wording That Changes the Model

A percentage must be applied to the base specified in the problem. “Increase by 5% of the preceding payment” is geometric. “Increase by an amount equal to 5% of the first payment each year” is arithmetic, because the dollar increment is fixed once the first payment is known. For a first payment of $800, the second wording adds $0.05(800)=$40$ each year, producing $800, $840, $880, and so on. The first wording produces $800, $840, $882, and so on.

Likewise, “increases by $30 and then by 2%” does not automatically specify a pure arithmetic or pure geometric pattern. If the rule applies both changes in each period, the next amount depends on the order and precise definition of those changes. Write the recurrence from the wording rather than forcing it into one of the two standard forms.

Exam trap: Applying 5% of the first payment repeatedly when the contract says “5% of the preceding payment” gives the wrong calculation. From $800, that mistaken method gives $800, 840, 880$, adding $40 each time. The correct geometric calculation is $800, 800(1.05)=840, 840(1.05)=882$. The second increase is $42 because it is 5% of $840, not 5% of $800.

When the Data Are Not Enough

A schedule with only two distinct, nonzero payments cannot by itself distinguish the two models. Given $C_1$ and $C_2$, an arithmetic rule can use $G=C_2-C_1$, while a geometric rule can use $r=C_2/C_1-1$. Both reproduce those two values. A third payment or an explicit rule is needed to test which pattern continues.

With three or more payments, compare all adjacent differences and ratios. A sequence may be neither type: for example, the differences might change and the ratios might also change. A finite sequence can occasionally fit both descriptions, particularly when all its payments are level; do not assume that the labels are always mutually exclusive.

This identification matters before any present-value or accumulated-value calculation. The payment sequence determines the amounts to be valued, and using a fixed-dollar gradient in place of percentage growth changes every later payment after the first increase. A reliable habit is to write the first few payments explicitly and check the rule against them before proceeding.

Check Your Understanding

For each problem, identify the payment pattern and write or extend the sequence as requested.

A hypothetical grant pays $2,400 at the end of year 1 and increases by $125 each year for four annual payments. Write the four payments and identify the pattern. (Answer: Arithmetic; $2,400, $2,525, $2,650, $2,775.)
A hypothetical subscription costs $640 for its first annual payment and rises by 7.5% of the preceding payment each year. Find the third payment. (Answer: Geometric; $640(1.075)^2=$$739.60.)
A schedule lists $900, $945, $992.25, and $1,041.8625. Test whether it is arithmetic or geometric, and give the rate if applicable. (Answer: Geometric with $r=5\%$; each consecutive ratio is 1.05.)
A hypothetical annual allowance begins at $1,500. Each year it increases by 4% of its first-year amount, not 4% of the prior year's amount. Write the first four payments and classify the pattern. (Answer: Arithmetic with $G=$60$; $1,500, $1,560, $1,620, $1,680.)
Three successive payments are $500, $550, and $610. Determine whether the three amounts form an arithmetic sequence, a geometric sequence, both, or neither. (Answer: Neither; the differences are $50 and $60, while the ratios are 1.10 and approximately 1.1091.)

What You'll Learn