What Is Arithmetic And Geometric Progression?
Arithmetic and geometric progression are the two most common types of number sequence, and they differ by one rule: how you get from each term to the next. In an arithmetic progression you add the same number every time. In a geometric progression you multiply by the same number every time.
An arithmetic progression (AP) is a list where consecutive terms differ by a fixed amount called the common difference, written $d$. The sequence $3, 7, 11, 15, \dots$ is an AP with $d = 4$, because each term is the one before it plus $4$.
A geometric progression (GP) is a list where consecutive terms have a fixed ratio called the common ratio, written $r$. The sequence $3, 6, 12, 24, \dots$ is a GP with $r = 2$, because each term is the one before it times $2$.
Throughout this article we use two running sequences:
$$\text{AP: } 3, 8, 13, 18, \dots \quad (a = 3,; d = 5) \qquad \text{GP: } 2, 6, 18, 54, \dots \quad (a = 2,; r = 3)$$
Here $a$ is the first term. From these two numbers and one rule, everything else about a progression follows.
How Do You Find The Nth Term Of An Arithmetic Progression?
To reach the nth term of an AP, start at the first term and add the common difference $(n-1)$ times, because the first term needs no addition, the second needs one, and so on.
$$a_n = a + (n - 1)d$$
Here $a_n$ is the nth term, $a$ is the first term, $d$ is the common difference, and $n$ is the position.
Example 1: Find the 10th term of the AP $3, 8, 13, 18, \dots$
With $a = 3$, $d = 5$, and $n = 10$:
$$a_{10} = 3 + (10 - 1)(5) = 3 + 45 = 48$$
Final answer: the 10th term is $48$. Notice the pattern grows in a straight line, every step is exactly $5$ higher than the last.
How Do You Find The Sum Of An Arithmetic Progression?
To add the first $n$ terms of an AP, pair the first term with the last, the second with the second-last, and so on. Every pair adds to the same total, which collapses the whole sum into one formula.
$$S_n = \frac{n}{2}\big[,2a + (n - 1)d,\big]$$
If you already know the last term $\ell$, the same sum is $S_n = \frac{n}{2}(a + \ell)$, first plus last, times how many terms, halved.
Example 2: Find the sum of the first 10 terms of the AP $3, 8, 13, 18, \dots$
With $a = 3$, $d = 5$, $n = 10$:
$$S_{10} = \frac{10}{2}\big[,2(3) + (10 - 1)(5),\big] = 5,[,6 + 45,] = 5 \times 51 = 255$$
Final answer: the first ten terms add to $255$. For more cases and a full derivation, see sum of n terms of an AP.
How Do You Find The Nth Term Of A Geometric Progression?
To reach the nth term of a GP, start at the first term and multiply by the common ratio $(n-1)$ times. Multiplying $(n-1)$ times is the same as raising $r$ to the power $(n-1)$.
$$a_n = a,r^{,n-1}$$
Here $a_n$ is the nth term, $a$ is the first term, $r$ is the common ratio, and $n$ is the position.
Example 3: Find the 5th term of the GP $2, 6, 18, 54, \dots$
With $a = 2$, $r = 3$, $n = 5$:
$$a_5 = 2 \times 3^{,5-1} = 2 \times 3^{4} = 2 \times 81 = 162$$
Final answer: the 5th term is $162$. Where the AP added $5$ each step, the GP triples, so it pulls ahead fast. A deeper walkthrough lives at nth term of a GP.
How Do You Find The Sum Of A Geometric Progression?
To add the first $n$ terms of a GP, use the formula below, which works whenever the common ratio is not $1$.
$$S_n = \frac{a,(r^{,n} - 1)}{r - 1}, \qquad r \neq 1$$
Example 4: Find the sum of the first 5 terms of the GP $2, 6, 18, 54, \dots$
With $a = 2$, $r = 3$, $n = 5$:
$$S_5 = \frac{2,(3^{5} - 1)}{3 - 1} = \frac{2,(243 - 1)}{2} = \frac{2 \times 242}{2} = 242$$
Final answer: the first five terms add to $242$. See sum of a GP for the reversed form used when $r < 1$.
Two edge cases decide whether the formula above even applies:
When $r = 1$, every term equals $a$, so the sum is simply $S_n = na$. The general formula would divide by $r - 1 = 0$, which is undefined, so this case is handled separately.
When $|r| < 1$, the terms shrink toward zero, and an infinite GP settles on a finite total: $S_\infty = \dfrac{a}{1 - r}$. This surprising result is covered in full at sum of an infinite GP.
What Is The Difference Between Arithmetic And Geometric Progression?
The two progressions share a skeleton, a first term and a repeated rule, but the rule is where they part ways. One table settles the whole comparison.
Table: Arithmetic progression versus geometric progression, dimension by dimension.
Dimension | Arithmetic Progression (AP) | Geometric Progression (GP) |
|---|---|---|
Rule between terms | Add a fixed number | Multiply by a fixed number |
Fixed quantity | Common difference $d = a_n - a_{n-1}$ | Common ratio $r = a_n / a_{n-1}$ |
nth term | $a_n = a + (n-1)d$ | $a_n = a,r^{,n-1}$ |
Sum of first $n$ terms | $S_n = \frac{n}{2}[2a + (n-1)d]$ | $S_n = \frac{a(r^{,n}-1)}{r-1},; r \neq 1$ |
Type of growth | Linear (straight line) | Exponential (curve) |
Example | $3, 8, 13, 18, \dots$ | $2, 6, 18, 54, \dots$ |
Infinite sum | Always diverges | Converges to $\frac{a}{1-r}$ when $\lvert r \rvert < 1$ |
Read the top two rows first. If consecutive terms differ by the same amount, it is an AP with linear growth; if consecutive terms divide to the same ratio, it is a GP with exponential growth. That one check tells you which set of formulas to use.
Why Do Arithmetic And Geometric Progressions Matter?
These two patterns are not just exercises. They are the mathematics of the two ways things change in the world.
Arithmetic progression models steady, linear change. A salary that rises by a fixed raise each year, a water tank filling at a constant rate, a taxi meter adding the same fare per kilometre. Whenever the amount added stays the same, an AP describes it.
Geometric progression models multiplying, exponential change. Money earning compound interest, a population that grows by a fixed percentage, a rumour that doubles its listeners each hour. Whenever the factor stays the same, a GP describes it.
The deep idea is the difference between adding and multiplying repeated over time. Adding the same amount keeps the growth flat and predictable. Multiplying by the same amount feeds each step back into the next, and small ratios build into enormous totals.
A GP with $r$ even slightly above $1$ will, given enough terms, overtake any AP no matter how large its common difference. That is the whole reason compound interest outruns simple interest, and why exponential ideas feel counterintuitive: our instincts are trained on arithmetic, but much of nature runs geometric.
Who Discovered Arithmetic And Geometric Progressions?
People were summing these sequences long before they had the notation we use today. The most famous story belongs to a bored schoolboy who found the AP sum formula in his head.
One other figure shaped the geometric side of the story:
Euclid of Alexandria (active around 300 BCE, Egypt) proved results about geometric progressions in Book IX of his Elements, including a formula for summing them, centuries before algebraic symbols existed. His work on geometric series still underlies how we sum a GP today.
Where Are Arithmetic And Geometric Progressions Used In The Real World?
The two patterns quietly run a wide range of everyday systems.
Personal finance: simple interest and fixed monthly savings form an AP, while compound interest and repeated percentage returns form a GP.
Biology and medicine: cell division, bacterial growth, and the spread of an infection early on follow a GP, doubling and redoubling.
Computer science: the running time of algorithms is often described by geometric sums, and evenly spaced loop counters step through an AP.
Music and physics: the frequencies of octaves double from one to the next, a GP, while evenly spaced markings on a ruler or a timeline form an AP.
Everyday planning: stadium seating that adds a fixed number of seats per row is an AP; a chain message that forwards to a fixed number of new people each round is a GP.
One pair of ideas, add-the-same or multiply-the-same, sits under money, biology, computing, and music alike. Recognising which one you are looking at is often the first step to solving a real problem.
What Are The Most Common Arithmetic And Geometric Progression Mistakes?
These three errors account for most lost marks on progressions, and each is documented in the comparison guides students read most (GeeksforGeeks, Cuemath, and the Voyager Sopris arithmetic-vs-geometric explainer).
Confusing adding with multiplying
Where it slips in:
A student sees $2, 6, 18, 54$ and calls it an AP with "common difference" $4$, reading the first gap ($6 - 2 = 4$) without checking the next.
Don't do this:
Do not decide the type from one gap. The gaps here are $4$, then $12$, then $36$, which are not equal, so it is not an AP.
The correct way:
Test both rules. Subtract consecutive terms for a common difference; divide consecutive terms for a common ratio. Here $6/2 = 18/6 = 54/18 = 3$, a constant ratio, so it is a GP with $r = 3$.
Using the AP formula on a GP (or the reverse)
Where it slips in:
A student correctly identifies a GP but then reaches for $a_n = a + (n-1)d$ out of habit, or applies $\frac{n}{2}[2a + (n-1)d]$ to sum it.
Don't do this:
Do not mix toolkits. An AP formula on a GP gives a straight-line answer for a curve, and it will be badly wrong.
The correct way:
Match the formula to the pattern. GP nth term is $a,r^{,n-1}$; GP sum is $\frac{a(r^{n}-1)}{r-1}$. Write the type at the top of your working before you pick a formula.
Dividing by zero in the GP sum when $r = 1$
Where it slips in:
A student applies $\frac{a(r^{n}-1)}{r-1}$ to a sequence like $7, 7, 7, 7$, where $r = 1$, and gets $\frac{0}{0}$.
Don't do this:
Do not use the general GP sum formula when the ratio is exactly $1$. The denominator $r - 1$ becomes $0$, which is undefined.
The correct way:
When $r = 1$, every term equals $a$, so the sum of $n$ terms is simply $S_n = na$. For $7, 7, 7, 7$ that is $4 \times 7 = 28$.
Practice Problems On Arithmetic And Geometric Progression
Answers follow each problem.
Is $5, 10, 20, 40, \dots$ an AP or a GP, and what is the fixed quantity?
(Answer: GP, common ratio $r = 2$.)Find the 12th term of the AP $4, 9, 14, \dots$
(Answer: $a_{12} = 4 + 11(5) = 59$.)Find the sum of the first 20 terms of the AP $2, 5, 8, \dots$
(Answer: $S_{20} = \frac{20}{2}[2(2) + 19(3)] = 10(61) = 610$.)Find the 6th term of the GP $3, 6, 12, \dots$
(Answer: $a_6 = 3 \times 2^{5} = 96$.)Find the sum of the first 5 terms of the GP $1, 4, 16, \dots$
(Answer: $S_5 = \frac{1(4^{5} - 1)}{4 - 1} = \frac{1023}{3} = 341$.)Find the sum to infinity of the GP $8, 4, 2, 1, \dots$
(Answer: $r = \tfrac{1}{2}$, so $S_\infty = \frac{8}{1 - \frac{1}{2}} = 16$.)
Where Should You Go Next After Arithmetic And Geometric Progression?
Progressions open straight into the wider study of patterns and change, and a few natural doors lead onward.
Introduction to progressions. Step back to the big picture of how sequences and series fit together before specialising.
Sum of an infinite GP. Follow the strangest result here, how infinitely many shrinking terms can add to a finite number.
Sequences. Broaden out to sequences that are neither arithmetic nor geometric, and the tools for spotting any pattern.
If your child is building these foundations, a live Bhanzu trainer teaches progressions starting from the "why" (linear versus exponential growth) in the Bhanzu algebra program.
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