Welcome to today's mathematics lesson. In this session, we explore one of the most elegant patterns in number sequences — the Geometric Progression. We will discover what makes a sequence geometric, how to find any term within it, and how to calculate the sum of its terms. Let us begin.
First, let us understand what we mean by a sequence. A sequence is simply an arrangement of numbers written in a definite order, following a specific rule. For example, consider the sequence 2, 6, 18, 54, and so on. Here, each term is multiplied by 3 to get the next term. Or take 1, negative one-half, one-fourth, negative one-eighth — here each term is multiplied by negative one-half to produce the next. When we connect these terms with plus or minus signs, we call it a series. Though technically distinct, we often use the words sequence and series interchangeably.
Now, what exactly is a Geometric Progression? A Geometric Progression, or G.P., is a sequence in which each term can be obtained by multiplying or dividing the preceding term by a fixed quantity. This fixed quantity is called the common ratio, denoted by the letter r.
Let us see this in action. Take the sequence 2, 4, 8, 16, continuing onward. Each term multiplied by 2 gives the next, so the common ratio r equals 2. Or consider 6, 2, two-thirds, and so on. Here, each term is multiplied by one-third to get the next, making r equal to one-third. Even sequences with negative terms and fractions follow this pattern — like 60, negative 24, 9 and three-fifths, negative 3 and 21 twenty-fifths, where r equals negative two-fifths.
To find the common ratio, simply divide any term by its preceding term. For the G.P. 16, 8, 4, 2, 1, one-half, and so on, we get 8 over 16 equals one-half, 4 over 8 equals one-half, 2 over 4 equals one-half. Since this ratio stays constant, we confirm it is a G.P. with r equal to one-half.
If you know the first term a and the common ratio r, you can write out the entire progression. Suppose a equals 5 and r equals 3. Then the G.P. becomes 5, then 5 times 3 which is 15, then 15 times 3 which is 45, then 45 times 3 which is 135, and so on indefinitely.
Now comes a crucial formula — the general term of a G.P.
Let the first term be a and the common ratio be r. Then the nᵗʰ term, which we call the general term, is given by arⁿ⁻¹. This means the first term is ar⁰ which equals a, the second term is ar¹ which equals ar, the third term is ar², and so on.
So the 8ᵗʰ term would be ar⁷, the 15ᵗʰ term would be ar¹⁴, and the 20ᵗʰ term would be ar¹⁹.
If a G.P. has exactly n terms, we call the nᵗʰ term the last term, denoted by l. Therefore, l = arⁿ⁻¹.
Here is a worked example. Find which term of the G.P. 3 minus 6 plus 12 minus 24 continuing onward equals negative 384. Here, a equals 3 and r equals negative 6 divided by 3, which is negative 2. Setting the nᵗʰ term equal to negative 384, we get 3 × (−2)ⁿ⁻¹ = −384. Dividing both sides by 3, we obtain (−2)ⁿ⁻¹ = −128. Since negative 128 equals negative 2 raised to the 7th power, we have n minus 1 equals 7, so n equals 8. Thus, negative 384 is the 8ᵗʰ term.
We can also find terms from the end of a G.P. The nᵗʰ term from the end equals l × (1/r)ⁿ⁻¹, where l is the last term. In our example, to find the 6ᵗʰ term from the end, we calculate −384 × (1/−2)⁵, which simplifies to 12.
Let us explore some fundamental properties of Geometric Progressions.
First, the ratio between consecutive terms always remains constant — this is the defining characteristic.
Second, three numbers a, b, and c are in G.P. if and only if b² = ac. This is a powerful test for geometric sequences.
Third, if you multiply or divide every term of a G.P. by the same non-zero number, the result is still a G.P.
Fourth, taking the reciprocals of all terms in a G.P. produces another G.P. For instance, if 6, 24, 96, 384 is a G.P., then one-sixth, one-twenty-fourth, one-ninety-sixth, one-three-hundred-eighty-fourth is also a G.P.
Fifth, raising every term to the same non-zero power yields another G.P. If 2, 8, 32 is a G.P., then 2 to the fifth power, 8 to the fifth power, 32 to the fifth power is also a G.P.
Sixth, if you multiply or divide corresponding terms of two different G.P.s, the resulting sequence is also a G.P. Suppose we have a, ar, ar squared, ar cubed and also A, AR, AR squared, AR cubed. Multiplying gives aA, ar times AR, ar squared times AR squared, and so on — clearly a G.P. with first term aA and common ratio rR.
Now we turn to one of the most important results — the sum of n terms of a G.P.
Let the first term be a, the common ratio be r, and the number of terms be n. The sum Sₙ equals a plus ar plus ar squared plus ar cubed, continuing up to arⁿ⁻¹.
We must consider three cases.
Case one: when r equals 1. Here, every term equals a, so the sum is simply n times a.
Case two: when the absolute value of r is less than 1. By multiplying the sum by r and subtracting from the original, we obtain Sₙ = a(1−rⁿ)/(1−r).
Case three: when the absolute value of r is greater than 1. We can rewrite the formula as Sₙ = a(rⁿ−1)/(r−1).
Notice that when the absolute value of r is less than 1, we use the form with 1 minus r to the n in the numerator and 1 minus r in the denominator. When the absolute value of r exceeds 1, we reverse both to r to the n minus 1 over r minus 1.
Let us work through an example. Find the sum of 10 terms of the series 96 minus 48 plus 24 and so on. The first term a is 96, and r equals negative 48 over 96, which is negative one-half. Since the absolute value of r is one-half, less than 1, we use Sₙ = a(1−rⁿ)/(1−r). Thus S₁₀ = 96[1−(−½)¹⁰]/(1−(−½)). This becomes 96 times two-thirds times 1023 over 1024, which simplifies to 63 and three-quarters.
Another example: find the sum of the G.P. 1, one-half, one-fourth, one-eighth, up to 12 terms. Here a equals 1, r equals one-half, and n equals 12. Using our formula, S₁₂ = 1×[1−(½)¹²]/(1−½). This gives 2 times 4095 over 4096, which equals 4095 over 2048, or 1 and 2047 over 2048.
When we know the last term rather than the number of terms, we can use an alternative formula: the sum equals (lr − a)/(r − 1) when r exceeds 1, or (a − lr)/(1−r) when r is less than 1.
Finally, let us discuss the Geometric Mean.
If a and b are two positive numbers, and G is the geometric mean between them, then a, G, b form a G.P. This means G over a equals b over G, giving G² = ab. Therefore, G = √ab.
For example, the geometric mean between 3 and 12 equals the square root of 3 times 12, which is the square root of 36, giving 6. Similarly, between 3 and 243, we get the square root of 729, which is 27.
Let us recap the key takeaways from today's lesson.
First, a Geometric Progression is a sequence where each term is found by multiplying the previous term by a fixed constant called the common ratio r.
Second, the nᵗʰ term of a G.P. is given by arⁿ⁻¹, where a is the first term.
Third, three numbers are in G.P. if and only if the square of the middle term equals the product of the first and third terms.
Fourth, the sum of n terms depends on whether the common ratio equals 1, has absolute value less than 1, or has absolute value greater than 1 — with specific formulas for each case.
Fifth, the geometric mean between two positive numbers a and b is the square root of their product.
And sixth, G.P.s preserve their structure under multiplication, division, reciprocals, and powers — making them remarkably stable mathematical objects.
Geometric Progressions appear throughout mathematics and its applications — in compound interest, population growth, radioactive decay, and countless natural phenomena. Mastering these concepts opens doors to understanding exponential processes in our world.
Thank you for your attention. Practice these formulas, work through derivations yourself, and you will find geometric progressions becoming second nature. Until next time, keep exploring the beautiful patterns of mathematics.