After learning the theory of arithmetic progression, we will see the basic fundas and applications of Geometric Progression. In AP, we used to get next term by adding a common difference to the previous term. But, in case of GP, we can obtain a new term by multiplying the previous term by a common ratio.
The first term is usually denoted by ‘a’ and the common ratio by ‘r’.
For e.g., 2, 4, 8, 16, 32,……………….is an example of GP with first term being 2 and common ratio being 2.
We can also say that 27, 9, 3, 1,…………………..is also an example of GP with a = 27 and r = 1/3.
As we found out the nth term of AP, we can also find out nth term of GP by the same logic.
So, t1 = a
Then, t2 = a × r, t3 = a × r × r or ar2, t4 = ar3, t5 = ar4,…………………..tn = a × r(n – 1)
So, we can use this formula whenever question is related with last term, first term or number of terms.
E.g. 1: Find the number of terms of the GP: 0.5, 1, 2,…………., 512.
This is a direct application of the formula learnt above, tn = a × r(n – 1).
Plug-in value of a = 0.5 and tn = 512 and r = 2, we get: 512 = 0.5 × 2n – 1
On solving, we get: 1024 = 2n – 1 which gives the value of n = 11. [1024 = 210.]
Sum to infinite terms of GP:
This is one special case of GP in which number of terms is infinite and the common ratio lies between –1 and 1. [–1 < r < +1].
E.g. 6: A rubber ball is thrown up in the air up to a height of 500 m, then it rebounds to a height of (4/5)th of the original. So, find the total distance traveled by the ball before it comes to rest.
This is a standard question based on the logic of sum of infinite terms of the GP.
If we take the first term as ‘a’ = 500. After the ball comes down and rebounds, distance travelled will be (500 × 4/5 = 400). The second time when it rebounds, the distance traveled by the ball will be (500 × 4/5 × 4/5), and the cycle will go on.
We do not need to go ahead, just think what all are required to find the sum of infinite terms. We need the first term and the common ratio ‘r’.
And both ‘a’ and ‘r’ are given in the question only. And i am saying that ‘r’ is already given in the question as the distance travelled after the first bounce is 4/5th of the original, so if you are smart enough you can identify there only that the common ratio will be 4/5. So, we don’t need to find out any other information.
Just plug-in the value in the formula of sum of infinite terms.
So, distance traveled will be = a/(1 – r), on solving, we get sum as 2500 m.
We will have to multiply this by 2 as we have not taken into account the distance which the ball travels when it comes down in the process. Every time the ball goes up after rebounding, it will come down, so we will have to multiply the distance by 2.
So, total distance travelled is 5000 m.
We just need to establish a relation between the sides of the outer-most square with the side of the immediate inner square. If we find the side of immediate Inner Square, we will get the area of that square and that will give us the common ratio of the infinite GP.
Side of the outer-most square is given as 10 units.
So, if the side of Inner Square is 5√2 units, the area of the inner square will be 50 sq. Units. And the area of the outer-most square is 100 square units. So, the area of the inner square has become 1/2 of the outer square. And the same relationship will also exist between the areas of Inner Square to its immediate inner square.
So, in all such geometrical problems, we just need to establish relation between the two immediate figures and same relation will also hold true for other immediate figures.
E.g. 11: Find the sum of the first 20 terms of the series:
4 + 44 + 444 + 4444 +………………………….. up till 20 terms.
This is a very random series and we should memorize this pattern as it becomes very difficult to identify which series is it.
We can rewrite this as 4[1 + 11 + 111 + 1111 +...............................].
E.g. 12: If four geometric means are inserted between 1/8 and 128. Find the third of this geometric mean.
This is a standard question based on GP. Do not get hassled by the term geometric mean, it is still based on the same logic.
If 4 geometric means are inserted between 1/8 and 128 means there are total of 6 terms in GP including the 4 means and the first term and the last term.
So, the series is like 1/8, ?, ?, ?, ?, 128.
So, we can assume the first term as a = 1/8, then the series becomes a, ar, ar2, ar3, ar4, ar5.
Now, think which data is given, a = 1/8 and ar5 = 128.
If we divide both of them, “a” will get canceled and we can find out the value of ‘r’.
On dividing, we get: (ar5)/a = (128) × 8 = 27 × 23 = 210.
So, we get r5 = 210, which gives us the value of r = 22.
We need to find the third geometric mean inserted means we need to find the value of ar3.
So, we know the first term ‘a’ = 1/8 and we know ‘r’ = 4, so ar3 = 1/8 × 43 = 8
So, the answer is 8.
Exercise:
1. On 1st of Jan 2006, two new societies P and Q are formed, each of ‘x’ members. On the first day of each subsequent month, P adds ‘a’ members while Q multiplies its current members by a constant multiplying factor ‘b’. It is observed that both the societies end up with same number of members on 2nd July, 2006. If a = 10.5x, then find the value of ‘b’.
2. In a GP, the first term is 7 and the nth term is 448. It is also given that the sum to the first ‘n’ terms is 889, and then determines the common ratio of this GP?
3. In the sequence 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8,………………….., find the 1035th term of the sequence.
4. The sum of an infinite GP is 4 and the sums of the cubes of the terms of the same GP are 192, and then find the common ratio of the original GP.
5. In a GP of even number of terms, the sum of all the terms is 5 times the sum of the odd terms. Find the common ratio of the GP.
6. The least value of ‘n’ for which the sum of 1 + 3 + 9 + 27 +…………..becomes greater than 3000 is
7. In an infinite GP, every term is equal to the sum of all the terms that follow. Find the common ratio.
8. Find the number of terms common between the series:
1 + 2 + 4 + 8 + 16 +…………………..50 terms.
4 + 7 + 10 + 13 + 16 +……………….90 terms.
9. The sequence Sn of positive terms is in GP such that the ratio of the 2nd term and 4th term is 1: 4. It is also given that the sum of the first term and the fourth term is 108. Then find the value of sixth term?
10. In a GP, the product of the first four terms is 4 and the second term is the reciprocal of the fourth term. The sum of the GP up to infinite terms is
11. A GP consists of 500 terms. The sum of the terms occupying the odd places is S while the sum of terms at even places is T. Then find the common ratio of the GP (in terms of S and T).
12. In an infinite GP, each term is equal to four times the sum of all the terms that follow. Find the common ratio.
13. If Y is the first term of an infinite GP and the sum to infinite terms of GP is 12. Then find the range of ‘Y’.
14. The middle-points of the sides of a equilateral triangle are joined to form a second equilateral triangle. Again a third equilateral triangle is constructed by joining the middle-points of this second triangle and the process is continued infinite times. If the area of the outermost equilateral triangle is Y square units, then find the sum of the areas of all such triangles.
15. If a and b are the roots of x2 – 15x + p = 0 and b and c are roots of x2 – 12x + q = 0. If a, b and c are in GP, then find the value of (p + q).
16. The sum of the first 10 terms of an AP is equal to 155 and sum of the first two terms of a GP is 9. How many values for the first term of an AP is possible if the first term of an AP is equal to the common ratio of the GP and the first term of the GP is same as the common difference of the AP?
17. The seventh term of a GP is 8 times the fourth term. What will be the first term when its fifth term is given as 48?
18. Determine the first term of a GP, if the sum of the first term and third term is 40 and the sum of the second and the fourth term is 80.
19. Find the sum to the first 20 terms of the series: 11 + 103 + 1005 + 10007 +………….
