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.

1. ARITHMETIC AND GEOMETRIC
PROGRESSIONS
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BBA 103 Business Mathematics
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2. Outline
• Introduction
• Sequence and Series
• Arithmetic Progression
• Arithmetic Means
• Geometric Progression
• Geometric Means
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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
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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
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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
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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)
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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, A1, A2, A3, ……., An, b
• An = a + nd (total terms n + 2)
=> An = a + n(b-a)
n + 1
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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
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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)
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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
 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))
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11. Harmonic Progression
HP:
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12. Harmonic Progression
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13. Summary
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14. Summary
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15. Summary
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