# INFINITE GEOMETRIC SEQUENCE HELP

A) determine whether the given sequence is geometric. If it is geometric find the common ratio.

{(5/4))^n}

B) Find the 7th term of the following geometric sequence

0.1,1.0,10.0,.........

C) Find the sum of the following infinite geometric sequence.

1-3/4+9/16-27/64+..................

### Comments for INFINITE GEOMETRIC SEQUENCE HELP

 Apr 21, 2011 Infinite Geometric Sequence by: s ----------------------------------------- Part II C) Find the sum of the following infinite geometric sequence. 1-3/4+9/16-27/64+.................. a_n = 1, -3/4, 9/16, -27/64, . . . r, the common ration is: r_n = a_3/a_2 = -27/64 ÷ 9/16 = -432/576 = -.75 the geometric sequence is: a_n = a_1 * (r)^(n - 1) a_n = 1 * (-.75)^(n - 1) this infinite geometric series is: S = 1 - ¾ + 9/16 - 27/64 + … an infinite geometric series will converge if: -1 < r < +1 r = -0.75; this series will converge (this series will converge because each consecutive term gets smaller and smaller) the general formula for the sum of an infinite geometric series is: S_∞ = a_1/(1 - r) The sum of this infinite geometric series is: S_∞ = 1/[1 - (-.75)] S_∞ = 1/(1 + .75) S_∞ = 1/(1.75) S_∞ = 0.571429 The final answer is: S_∞ = 0.571429 Thanks for writing. Staff www.solving-math-problems.com

 Apr 21, 2011 Infinite Geometric Sequence by: Staff Part I The question: A) determine whether the given sequence is geometric. If it is geometric find the common ratio. {(5/4))^n} B) Find the 7th term of the following geometric sequence 0.1,1.0,10.0,......... C) Find the sum of the following infinite geometric sequence. 1-3/4+9/16-27/64+.................. The answer: A) determine whether the given sequence is geometric. If it is geometric find the common ratio. {(5/4))^n} There is an extra right parenthesis in {(5/4))^n}. I think this is what you mean: a_n = (5/4)^n the first four terms in the sequence are: if n = 1, the first term is a_1 = (5/4)¹ a_1 = 5/4 if n = 2, the second term is a_2 = (5/4)² a_2 = 5²/4² a_2 = 25/16 if n = 3, the third term is a_3 = (5/4)³ a_3 = 5³/4³ a_3 = 125/64 if n = 4, the fourth term is a_4 = (5/4)⁴ a_4 = 5⁴/4⁴ a_4 = 625/256 a_n = 5/4, 25/16, 125/64, 625/256, . . . “If” this is a geometric sequence, then r, the common ratio, will be the same no matter what two consecutive terms are used to calculate the ratio: r_n = a_n/a_n-1 (n must be greater than 1) r_n = r with a subscript of n (this is the common ratio) a_n = a with a subscript of n (this is the nth term in the sequence) a_n-1 = a with a subscript of n-1 (this is the n-1 term in the sequence) using the 1st and 2nd terms in the sequence to calculate the common ratio r_n = a_2/a_1 25/16 ÷ 5/4 = 100/80 = 5/4 = 1.25 using the 2nd and 3rd terms in the sequence to calculate the common ratio r_n = a_3/a_2 125/64 ÷ 25/16= 2000/1600 = 5/4 = 1.25 using the 3rd and 4th terms in the sequence to calculate the common ratio r_n = a_4/a_3 625/256 ÷ 125/64 = 40000/32000 = 5/4 = 1.25 The calculated ratios all have the same value: r_n = a_2/a_1 = a_3/a_2 = a_4/a_3 = 5/4 = 1.25 Therefore, the sequence a_n = (5/4)ⁿ is a geometric sequence B) Find the 7th term of the following geometric sequence 0.1,1.0,10.0,......... using the 2nd and 3rd terms in the sequence to calculate the common ratio r_n = a_3/a_2 10.0 ÷ 1.0 = 10 The common ratio, r, is equal to 10. A geometric sequence has the (general) form: a_n = a_1 * (r)^(n - 1) a_n = a with a subscript of n (this is the nth term in the sequence) a_1 = a with a subscript of 1 (this is the 1st term in the sequence) n = number of terms r = the common ratio The equation for this geometric sequence is: a_1 = 0.1 r = 10 a_n = a_1 * (r)^(n - 1) a_n = 0.1 * (10)^(n - 1) The 7th term in the sequence is: a_7 = 0.1 * (10)^(7 - 1) a_7 = 0.1 * (10)⁶ a_7 = (10)⁵ a_7 = 100,000 -----------------------------------------

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