Arithmetic Progression (A.P) and Geometric Progression (G.P)

 

Arithmetic Progression (A.P)

An AP is a sequence in which the terms increase or decrease by a constant number, called Common Difference.

Let a, a+d, a+2d, a+3d …………… are said to be in A.P, in which

            1^st term = a, Common Difference = d  & n= no. of terms  

1.      nth term or last term = t_n = a + (n-1)d

2.      Sum of ‘n’ terms = S_n = n/2 [2a + (n-1)d] = n/2 [a + t_n]

Examples

1.     1. Find the sum of the first 20 terms of the AP; 5,2,-1,-4,-7………..

Solution:

                    Here a= 5, d=2-5 = -3, n=20

                   S_n = n/2 [2a + (n-1)d] = 20/2 [2 × 5 + (20-1)(-3)] = -470

 2.   How many natural numbers are there between 23 and 100 which are exactly divisible by 6?

Solution:

          The nos. are 24, 30,36,…………………..96

          Here, a = 24, d = 6 & t_n = 96

                Then n= ?

           As t_n = a + (n-1)d = 96

                 ⟹ 24 + (n-1) × 6 = 96

                 ⟹ (n-1) × 6 = 72

                 ⟹ n = 13 (Ans.)

 3.  6 + 15 + 24 + 33 + ……….. + 105 = ?

Solution:

           Here common difference = d = 9 , so the series is in AP.

                    a = 6 , t_n = last term = 105

                     so, n = ?

                      t_n = a + (n-1)d = 105

                      ⟹ 6 + (n-1) × 9 = 105

                      ⟹ (n-1) × 9 = 99

                     ⟹ n = 12

 Sum = n/2 [a + t_n] = 12/2 [6 + 105] = 666 (Ans.)

 Geometric Progression (G.P)

A GP is a sequence in which the terms increase or decrease by a constant ratio, called Common Ratio.

Let a, ar, ar², ar³ …………… are said to be in G.P, in which

            1^st term = a, Common Ratio = r & n= no. of terms   

1.      nth term or last term = t_n = arⁿ⁻¹

2.      Sum of ‘n’ terms = S_n  is given by

                                     S_n = a (1-rⁿ)/(1-r) , where r < 1

                                     S_n = a (rⁿ -1)/(r-1) , where r > 1

•  For infinite series of G.P, the required sum = a / (1-r)

 

Examples

1.     1. How many terms are there in G.P 3,6,12,24,………..,384

   Solution:

                    Here a= 3, r=6/3 = 2, n= ?

                    last term = t_n = arⁿ⁻¹ = 3 × 2ⁿ⁻¹ = 384

                                                   ⟹ 2ⁿ⁻¹ = 128 = 2⁷

                                                   ⟹ n – 1 = 7 

                                                   ⟹ n = 8 (Ans.)

 2. 2 + 2² + 2³ +2⁴ +……………+ 2⁹ = ?

Solution:

            Here r = 2 , a = 2 , n =9

            Then, S_n = a (rⁿ -1)/(r-1)

                        = 2 (2⁹ - 1)/ (2-1) = 1022 (Ans.)

Harmonic Progression (H.P)

A sequence is said to be in H.P, if the reciprocal of the terms are in A.P.

If  ‘a’ & ‘b’ are said to be in H.P, when 1/a and 1/b are in A.P.

Arithmetic Mean and Geometric Mean

Arithmetic Mean (AM)	Geometric Mean  (GM)
If a, b, c are in AP, then b = (a + c)/2 where b is called the AM of series a, b, c. Similarly, for n terms series,  AM = (a₁ + a₂ + a₃ +…………+ aₙ) / n	If a, b, c are in GP, then b =√(ac) where b is called the GM of series a, b, c. Similarly, for n terms series, GM = (a₁ × a₂ × a₃ ×…………× aₙ)¹/ⁿ

Do it yourself

1. If the first term of a G.P. is 20 and the common ratio is 4. Find the 5th term. (Ans.5120)
2.  The sum of the first three terms of a G.P. is 21/2 and their product is 27. Find the common ratio. (Ans. 2 &1/2)
3.  If the nth term of a GP is 128 and both the first term a and the common ratio r are 2. Find the number of terms in the GP. (Ans. n=7)
4. What is the sum of the following series?   -64, -66, -68, ..... , -100.
5. Find the first term of the AP series in which 10^th term is 6 and 18^th term is 70. (Ans. -66).
6.  Find the n^th term of the series 3, 8, 13, 18,….. (Ans. 5n-2)
7. Find the sum of the following infinite G. P. 1/3,1/9,1/27,1/81, ​……. (Ans. 1/2)
8. What is the sum of infinite geometric series with first term equal to 1 and common ratio is ½? (Ans.2)
9. What is the sum of 6 + 7 + 8 + ........... + 16 ? (Ans.121)
10. What is sum total of all the figures from 31 to 50? (Ans.810) 
11. What is the sum of the first 17 terms of an arithmetic progression if the first term is -20 and last term is 28 ? (Ans. 68)
12. If A = 1 - 10 + 3 - 12 + 5 - 14 + 7 ....... upto 60 terms, then what is the value of A?
13. How many terms are there in an AP whose first and fifth terms are -14 and 2 respectively and the sum of terms is 40 ? (Ans. 10)