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![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…………..
• an = a + (n-1)d
• Sum of First n Terms of an A.P.
• Sn = n [2a + (n - 1)d]
2
8/29/2019 BBA 103 Business Mathematics 5](https://image.slidesharecdn.com/4-190829044305/85/4-ap-gp-5-320.jpg)
Uploaded byPSIT kanpur
4. ap gp
This document discusses arithmetic and geometric progressions. It defines arithmetic and geometric sequences as lists of numbers where each subsequent term is calculated using a common difference or ratio. It provides formulas to calculate the nth term and sum of the first n terms for both progressions. The document also discusses arithmetic and geometric means as the averages between two numbers in an arithmetic or geometric progression.
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4. ap gp
- 1. ARITHMETIC AND GEOMETRIC PROGRESSIONS 8/29/2019 BBA103 Business Mathematics 1
- 2. Outline • Introduction • Sequenceand Series • Arithmetic Progression • Arithmetic Means • Geometric Progression • Geometric Means 8/29/2019 BBA 103 Business Mathematics 2
- 3. What You ShouldLearn • Understand the concept of sequence and series • Identify arithmetic and geometric progressions • Apply A.P. and G.P. in various business problems 8/29/2019 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 • Sn represents the sum of the first n terms • S1 = a1 • S2 = a2 • Sn = a1 + a2 + a3 + ……….. an 8/29/2019 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………….. • an = a + (n-1)d • Sum of First n Terms of an A.P. • Sn = n [2a + (n - 1)d] 2 8/29/2019 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) 8/29/2019 BBA 103 Business Mathematics 6
- 7. Arithmetic Means • Anumber (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, A1, A2, A3, ……., An, b • An = a + nd (total terms n + 2) => An = a + n(b-a) n + 1 8/29/2019 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, ar2, ar3………….. • an = arn - 1 • Sum of First n Terms of an A.P. • Sn = a (rn – 1) (Assuming r is not equal to 1) r - 1 8/29/2019 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 = 22 = 4) 8/29/2019 BBA 103 Business Mathematics 9
- 10. Geometric Means • Anumber (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 G2 = ab G = +√ab • n geometric means between 2 numbers ‘a’ and ‘b’ • a, G1, G2, G3, ……., Gn, b • Gn = arn (total terms n + 2) Gn = a (b/a)(n/(n+1)) 8/29/2019 BBA 103 Business Mathematics 10
- 11. Harmonic Progression HP: 8/29/2019 BBA103 Business Mathematics 11
- 12. Harmonic Progression 8/29/2019 BBA103 Business Mathematics 12
- 13. Summary 8/29/2019 BBA 103Business Mathematics 13
- 14. Summary 8/29/2019 BBA 103Business Mathematics 14
- 15. Summary 8/29/2019 BBA 103Business Mathematics 15