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Example 13

and. Example 13. The first two terms of a geometric progression are 3 and –2. Find the least value of n for which the difference between sum of the first n terms and sum to infinity is within 2% of the sum to infinity. Since. Hence the least value of n is 10. (ans).

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Example 13

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  1. and Example 13 The first two terms of a geometric progression are 3 and –2. Find the least value of n for which the difference between sum of the first n terms and sum to infinity is within 2% of the sum to infinity.

  2. Since Hence the least value of n is 10. (ans)

  3. Given . Example 14 The sum of the first n terms of a series is Find the fifth term. Prove that the series is arithmetic and state its common difference. Solution Therefore,

  4. Given , To prove that the series is an A.P., prove

  5. Thus, the series is arithmetic with common difference 4.

  6. Example 15 The sum of the first n terms of a series is given by . By finding an expression for the nth term of the series, show that this is a geometric series, and state the value of the first term and the common ratio. Solution

  7. To show G.P., show

  8. Therefore, the series is geometric with first term and common ratio

  9. Example 16 The sum of the first 100 terms of an Arithmetic Progression is 10, 000; the first, second and fifth terms of this progression are three consecutive terms of a Geometric Progression. Find the first term, a , and the non-zero common difference, d, of the A.P. Solution ----(1) are three consecutive terms of a Geometric Progression a, a+d, a+ 4d

  10. ----(2) Substituting (2) into (1), From (2),

  11. The rth term of a series is . Find the sum of the first n terms. Example 17 Solution Given The sum of the first n terms is:

  12. GP: a = 1, r = 3, n terms AP: a = 1, d = 1, n terms

  13. Example 18 Each time that a ball falls vertically on to a horizontal floor it rebounds to three-quarters of the height from which it fell. It is initially dropped from a point 4 m above the floor. Find, and simplify, an expression for the total distance the ball travels until it is about to touch the floor for the (n+1)th time. Hence find the number of times the ball has bounced when it has traveled 24 m and also the total distance it travels before coming to rest. (The dimensions of the ball are to be ignored.)

  14. 4m 2nd 1st 3rd nth (n+1)th Total distance (in metres) that the ball travels

  15. When the ball has traveled 24 m, Hence find the number of times the ball has bounced when it has traveled 24 m and also the total distance it travels before coming to rest Let the number of times the ball has bounced be n. Solution

  16. For the ball to come to rest, Since n is an integer, least n=7. Therefore, the ball has bounced 7 times when it has traveled 24 m. The ball travels 28 m before coming to rest.

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