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| Geometric Progressions (G.P.) |
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| The series of terms a, ar, ar2, ar3,.... in which each term bears a constant ratio to the preceeding term is a geometric progression. The constant ratio is called the common ratio. |
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| A geometrical progression is a succession of terms such that each term bears fixed ratio to the preceeding term (i.e., is obtained by multiplying the preceeding term by a fixed quantity). The fixed ratio is called the common ratio of the geometric progression. |
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| Examples: |
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| From the two examples it is seen that the signs of the terms of a GP must either be all alike or alternatively positive and negative. |
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Note that the numbers in continued proportion are in GP, i.e.,  |
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| To find the nth term of a GP, whose first term is a common ratio r and number of terms is n |
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| We observe that the index of r on the right hand side is one less than the suffix of t on the left hand side in each of the equalities. Hence tn = arn-1 which is the general term of the given GP. |
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| An important note: |
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| If the product of three numbers in GP is given, take the term as
a/r, a, ar. But if the product of the numbers is not given, the terms are in the ordinary form. |
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| To find the sum of n terms of a GP |
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| Let a = First term, r = common ratio, n = number of terms. |
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| Multiply both sides of (i) by r, the common ratio. |
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| Subtracting (ii) from (i), we get |
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| To find the sum to infinity of a GP when the common ratio r is numerically less than 1 |
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| Consider the GP a, ar, ar2... |
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| Note: |
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| Sum to infinity exists only when r is numerically less than 1. i.e. |r|<1 |
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