Ever noticed how some number sequences multiply by the same value every time? That’s exactly what a Geometric Progression (GP) is all about! In a GP, each term keeps a constant ratio with the term before it—super handy when tackling pattern questions. Whether you're prepping for the SAT, ACT, PSAT/NMSQT, GED, or even advanced exams like GRE, GMAT, AP tests, PERT, Accuplacer, and MCAT, knowing the GP formulas will definitely give you an edge.

In this guide, we'll break down everything you need: the general formula for geometric progression, how to find the sum of the nth term, and the key properties you can rely on. We’ll also throw in easy-to-follow examples to help solidify your understanding. Trust us, mastering GPs not only helps in math sections but also boosts your confidence when dealing with series, patterns, and statistics-based questions across these major exams!

Geometric Progression

The type of progression where the next term is received by multiplying a fixed term (which is also known as the common ratio) every time to the preceding term is the geometric progression definition. Hence as per the definition, you can point out that in a GP:

• The sequence consists of non-zero numbers.
• The ratio of the term and its preceding term remains constant.
• The constant ratio is also known as a common ratio (r).

Mathematically we can state that; a sequence \(a_1, a_2, a_3, a_4, a_5, a_6, \dots\dots a_n\) is called a GP if \(\frac{a_{n+1}}{a_n} \)= constant.

Geometric Progression Formulas

Check the important geometric progression formulas below:

General Form of Geometric Progression

A Geometric Progression in mathematics can be finite or infinite depending on the given number of elements. That is 4, 16, 64, 256… is an infinite geometric progression example having a common ratio of 4. The general form of geometric progression(GP) is as follows:

\(a,ar,ar^2,ar^3,ar^4,\dots,\) where a denotes the first term and r is the common ratio.

Types of Geometric Progression

As of now, we can say that the geometric progression meaning is that you can locate all the terms of a GP, by just having the first term and the constant ratio. Now moving toward the types, there are two types of a GP.

• Finite Geometric Progression
• Infinite Geometric Progression

Let us learn about each of them one by one.

Finite Geometric Progression

In the previous heading we learn that the general form of a GP is ; \(a,ar,ar^2,ar^3,ar^4,\dots,\). Now as per the header a finite GP would include a finite number of terms. That is, in this particular type of progression the last term is specified. The general form for this pattern is: \(a, ar, ar^2, ar^3,……ar^{n-1}\).

Infinite Geometric Progression

In the previous header, you saw the formula for finite GP, let us learn about the infinite GP formula with definition. In the infinite geometric progression, an infinite number of terms are present. Or one can understand this as a progression where the last term is not specified. The general form for this pattern is: \(a, ar, ar^2, ar^3, \cdots ar^{n-1},\cdots\cdots\).

Sum of n Terms of a GP

At this point of our discussion we know that \(a, ar, ar^2, ar^3,……ar^{n-1}\) denotes a finite GP and \(a, ar, ar^2, ar^3, \cdots ar^{n-1},\cdots\cdots\) denotes an infinite GP. Moving towards the sum of geometric progression formulas.

The sum of n terms of G.P. whose first term is ‘a’ and the common ratio is ‘r’ depends on the following conditions:

• When |r|>1, then; \(S_n=a\left(\frac{r^n-1}{r-1}\right)\).
• When |r|<1, then; \(S_n=a\left(\frac{1-r^n}{1-r}\right)\).
• When r = 1, then; \(S_{n}=na\).

Also, the sum of the terms of the G.P. series when the number of terms in it is infinite is given by:

\(S_n=\frac{a}{1-r}\)

Properties of Geometric Progression (GP)

Now that you know the general form, finite and infinite GP representation along with the formula for the sum of n terms. Let us now learn some important properties related to the topic.

• When each term of a G.P. is multiplied and divided by a fixed non-zero number, then the resulting sequence is also a G.P.
• If there is two Geometric Progression a₁, a₂,  a₃, a₄, a₅, a₆, ………………, a_n and b₁, b₂,  b₃, b₄, b₅, b₆, ………………, b_n. Then,  a₁b₁, a₂b₂,  a₃b₃, a₄b₄, a₅b₅, a₆b₆, ………………, a_nb_n is also a geometric  Progression.
• Let there be three numbers a, b, and c. They are in G.P. if and only if \(b^{2}=ac\).
• The reciprocal of all the components in G.P are also in G.P.
• Similarly, if each term of a G.P is raised to the identical power, then the new sequences and series will also be in G.P.
• For the terms in the finite GP, the product of the terms equidistant from the opening and the ending is the same.

Solved Examples on GP

Now that you know the details regarding the definition, GP sum to infinity, the sum of n terms with detailed properties and related things. Let us proceed toward some solved GP problems to understand these things more clearly.

Solved Example 1:Check if the given sequence 8, 4, 2, 1, 1/2, 1/4…… is in GP or not.

Solution: To find the GP, let us check for the common difference between the terms. Here, \(a_1\)= 8, \(a_2\) = 4, \(a_3\)= 2, \(a_4\) = 1, \(a_5\) = 1/2 and \(a_6\) = 1/4.

The ratio of the consecutive terms is;

\(\frac{a_2}{a_1}=\frac{4}{8}=\frac{1}{2},\frac{a_3}{a_2}=\frac{2}{4}=\frac{1}{2},\ \frac{a_4}{a_3}=\frac{1}{2}=\frac{1}{2}\)

As, the ratio of the consecutive terms of the assigned sequence is 1/2, which is a fixed number, therefore, the given sequence is in GP.

Solved Example 2: Find the next terms of the provided geometric progression: 1,3,9,27,81,…

Solution: Given GP is 1,3,9,27,81,…

As per the definition, GP is a series of numerals wherein each term is calculated by multiplying the earlier term by the common ratio(a fixed number).

The common ratio is:

\(\frac{3}{1}=\frac{9}{3}=\frac{27}{9}\ =3\)

Hence we can say that 3 is the common ratio of the given series.

To check for the rato thing, 1,3,9,27,81,…

If we multiply 1 by 3 and we get 3. That is the second term of the given series.

Next, by multiplying the 3 with 3 we get 9 as the third term and so on. Therefore the common ratio is 3. So the next term of the series is 81 times 3 i.e. is equal to 243.

Solved Example 3: If the first term for a GP is 6 and the common ratio of the GP is 2. Then compose the first 4 terms of GP.

Solution: Given,

The first-term i.e. a = 6

Common ratio, i.e. r = 2

As per the formula, the general form of GP for the first 4 terms is given by:

\(a, ar, ar^2, ar^3\)

Substituting the values we get;

a =6

ar = 6 x 2 = 12

\(ar^2 = 6\times 2^2\) = 24

\(ar^3 = 6\times 2^3\) =48

Thus, the first 4 terms of GP starting with 6 as the first term and 2 as the common ratio is 6, 12, 24, 48.

Solved Example 4: Determine the sum of for the given GP; 5, 10, 20, 40, 80 using the sum of n terms formula.

Solution: Given GP is 5, 10, 20, 40, 80

a(first term)= 5

common ratio, r = 10/5 = 2

Number of terms, n = 5

The Sum of GP is given by the formula;

\(S_n=a\left(\frac{r^n-1}{r-1}\right)\), when |r|>1.

\(S_5=5\left(\frac{2^5-1}{2-1}\right)\)

\(=5\left(\frac{32-1}{2-1}\right)\)

\(S_5=5\left(\frac{31}{1}\right)=155\)

Cross check the given data: 5+10+20+ 40+ 80 = 155.

Conclusion

And there you have it! Geometric Progression (GP) might sound fancy, but it’s really just about spotting patterns where each term multiplies by the same number. Whether you're facing questions on the SAT, ACT, PSAT/NMSQT, GED, GRE, or any major test, knowing how to handle GPs—finding terms, sums, or spotting properties—can seriously boost your math game. Keep practicing, and these patterns will become second nature!