4. ap gp

Concept of AP & GP

1. Arithmetic and Geometric Progressions 5/28/2018 BBA 103 Business Mathematics 1

2. Outline Introduction Sequence and Series Arithmetic Progression Arithmetic Means Geometric Progression Geometric Means 5/28/2018 BBA 103 Business Mathematics 2

3. What You Should Learn Understand the concept of sequence and series Identify arithmetic and geometric progressions Apply A.P. and G.P. in various business problems 5/28/2018 BBA 103 Business Mathematics 3

4. Sequence and Series Sequence A list of numbers, called terms Always arranged in a definite order Each number, except first, follows another number according to some rule. Examples: 1,3,5,7,9 …… .. 2,4,6,8 (finite sequence) Series The indicated sum of the terms of a sequence is called an infinite series S n represents the sum of the first n terms S 1 = a 1 S 2 = a 2 S n = a 1 + a 2 + a 3 + ……… .. a n 5/28/2018 BBA 103 Business Mathematics 4

5. Arithmetic Progression (A.P.) Sequence of terms Each term after the first (a) equals the sum of the preceding term and a constant, called common difference (d) 1, 3, 5, 7 (d=2) 4, 7, 10, 13 (d=3) nth term of an A.P. a, a + d, a + 2d, a + 3d ………… .. a n = a + (n-1)d Sum of First n Terms of an A.P . S n = n [ 2a + (n - 1)d ] 2 5/28/2018 BBA 103 Business Mathematics 5

6. Properties of A.P. If a constant is added or subtracted from each term of an A.P., the resulting progression will also be in A.P. having the same common difference 1,3,5,7, … .. (d = 2) 1 is added to each term – 2,4,6,8, …… (d = 2) If each term of an A.P. is multiplied by a constant, the resulting progression will also be in A.P. having the same common difference multiplied by that constant 1,3,5,7, … .. (d = 2) 3 is multiplied by each term – 3,9,15,21, …… (d = 2*3 = 6) If each term of an A.P. is divided by a constant, the resulting progression will also be in A.P. having the same common difference divided by that constant 2,4,6,8, … .. (d = 2) each term is divided by 2 – 1,2,3,4, …… (d = 2/2 = 1) 5/28/2018 BBA 103 Business Mathematics 6

7. Arithmetic Means A number (A) is called an Arithmetic Means (A.M.) of two numbers (a and b) if all of them are in A.P. Since a, A, b are in A.P. A – a = b – A 2A = a + b A = a + b 2 n arithmetic means between 2 numbers ‘a’ and ‘b’ a, A 1 , A 2 , A 3 , …… ., A n , b A n = a + nd (total terms n + 2) => A n = a + n(b-a) n + 1 5/28/2018 BBA 103 Business Mathematics 7

8. Geometric Progression (G.P.) Sequence of terms Each term after the first (a) is formed by multiplying the preceding term by a constant, called common ratio ( r ) 2, 4, 8, 16 … . (r = 2) 3, 9, 27, 81 … .. (r = 3) nth term of an G.P. a, ar , ar 2 , ar 3 ………… .. a n = ar n - 1 Sum of First n Terms of an A.P . S n = a ( r n – 1) (Assuming r is not equal to 1) r - 1 5/28/2018 BBA 103 Business Mathematics 8

9. Properties of G.P. If a sequence is in G.P., then the sequence obtained by multiplying each terms by a non-zero constant is also in G.P. 2, 4, 8, 16, …… (r = 2) Each term is multiplied by 3 – 6, 12, 24, 48, …… (r = 2) If a sequence is in G.P., then the sequence obtained by taking the reciprocal of each term is also in G.P . 2, 4, 8, 16, …… (r = 2) Each term is reciprocated – ½, ¼, 1/8, 1/16 … .(r = ½) If a sequence is in G.P., then the sequence obtained by raising each term to the same power is also in G.P . 2, 4, 8, 16, …… (r = 2) each term is raised by 2 – 4, 16, 64, 256, …… (r = 2 2 = 4) 5/28/2018 BBA 103 Business Mathematics 9

10. Geometric Means A number (G) is called a Geometric Mean (G.M.) of two numbers (a and b) if all of them are in G.P. Since a, G, b are in G.P. G = b a G G 2 = ab G = + √ ab n geometric means between 2 numbers ‘a’ and ‘b’ a, G 1 , G 2 , G 3 , …… ., G n , b G n = ar n (total terms n + 2) G n = a (b/a) (n/(n+1)) 5/28/2018 BBA 103 Business Mathematics 10

11. Harmonic Progression HP: 5/28/2018 BBA 103 Business Mathematics 11

12. Harmonic Progression 5/28/2018 BBA 103 Business Mathematics 12

13. Summary 5/28/2018 BBA 103 Business Mathematics 13

14. Summary 5/28/2018 BBA 103 Business Mathematics 14

15. Summary 5/28/2018 BBA 103 Business Mathematics 15