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Questions and Answers
What is the formula for the nth term of an Arithmetic Progression (A.P.)?
In the sequence 2, 5, 8, 11, which of the following represents the first term and common difference?
For a geometric progression (G.P.), what is the necessary condition for three numbers a, ar, and ar² to be in G.P.?
What is the formula for the sum of the first n terms in a geometric progression (G.P.) when the common ratio r is not equal to 1?
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Which of the following correctly describes the series of the first n natural numbers using sigma notation?
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In the context of A.P., what does the symbol d represent?
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If the second and third terms of an A.P. are given as a+d and a+2d, how can you express the first term?
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What is the geometric mean of two numbers a and b?
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If you insert n arithmetic means between two numbers a and b, how is the common ratio r determined?
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Study Notes
Arithmetic Progression (A.P)
- The general term (nth term) of an A.P. is given by tn = a + (n-1)d, where 'a' is the first term and 'd' is the common difference.
- The sum of the first 'n' terms of an A.P is given by Sn = (2a + (n-1)d)/2.
- In an A.P., the difference between any two consecutive terms is constant, which is called the common difference.
- To determine if a sequence is an A.P., check if the difference between consecutive terms is constant.
- An A.P can be represented as a sequence with a constant difference between terms. Examples include: 2k, 5k, 8k, 11k or 2/k, 5/k, 8/k, 11/k or 2+k, 5+k, 8+k, 11+k or 2-k, 5-k, 8-k, 11-k.
- For a 3 term A.P., they can be expressed as: a, a+d, a+2d or a-d, a, a+d.
- For a 4 term A.P., they can be expressed as: a-3d, a-d, a+d, a+3d.
- For a 5 term A.P., they can be expressed as: a-2d, a-d, a, a+d, a+2d.
- In an A.P., the difference between the last term and the second last term is equal to the common difference.
- The difference between the nth term and (n-1)th term is equal to the common difference 'd', indicating the constant difference between terms.
Inserting Arithmetic Means
- We can insert 'n' arithmetic means between any two numbers 'a' and 'b' to form an A.P.
- The common difference (d) for this series is calculated as d = (b+a)/(n-1)
- The 'n' arithmetic means are: a + d, a + 2d, ..., a + nd
- The sum of all arithmetic means between 'a' and 'b' can be calculated as Sam = n(a+b)/2, where 'n' is the number of arithmetic means.
Geometric Progression (G.P)
- The general term (nth term) of a G.P. is given by tn = arn-1, where 'a' is the first term and 'r' is the common ratio.
- The sum of the first 'n' terms of a G.P is given by Sn = a(1-rn)/(1-r).
- If 'r' = 1, the sum of a G.P is given by Sn = na.
- The sum of an infinite G.P. for -1 < r < 1 is S0 = a/(1-r), where 'a' is the first term and 'r' is the common ratio.
- In a G.P, the ratio between any two consecutive terms is constant, which is called the common ratio. To determine if a sequence is a G.P, check if the ratio between consecutive terms is constant .
- A G.P can be represented as a sequence with a constant ratio between consecutive terms.
- For a 3 term G.P, they can be expressed as: a, ar, ar2 or a, a, ar
- For a 4 term G.P, they can be expressed as: a/r, a, ar, ar3
- For a 5 term G.P, they can be expressed as: a/r2, a/r, a, ar, ar2
- In G.P, the ratio between the nth term and (n-1)th term is equal to the common ratio.
- The difference between Sn and Sn-1 is equal to the nth term (Tn) of the G.P. Notably, Tn is equal to r (common ratio) multiplied by Tn-1 (previous term).
- In any G.P, if a, b, and c are consecutive terms, then b2 = ac. This can be further written as a/b = b/c.
Geometric Mean (GM)
- The geometric mean for two numbers, 'a' and 'b', is given by (ab)1/2.
- The geometric mean for three numbers, 'a', 'b', and 'c', is given by (abc)1/3.
- The geometric mean for 'n' numbers (a, b, c, ..., an) is given by (a, b, c,..., an)1/n.
Inserting Geometric Means
- We can insert 'n' geometric means between any two positive numbers 'a' and 'b' to form a G.P.
- The common ratio (r) for this series is calculated as r =( b/a)1/n.
A.M. and G.M. Inequality
- The arithmetic mean (AM) of a set of numbers is always greater than or equal to the geometric mean (GM) of the same set of numbers. This inequality can be expressed as A.M ≥ G.M.
Sigma Notation
- The following are some important formulas involving sigma notation:
- Σr=1n r = n(n+1)/2
- Σr=1n r2 = n(n+1)(2n+1)/6
- Σr=1n r3 = (n(n+1)/2) 2
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Description
This quiz covers the fundamentals of Arithmetic Progression (A.P), including the general term, sum of terms, and identifying A.P sequences. You will learn to apply the formulas and concepts related to A.P, enhancing your understanding of this important mathematical topic.