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The second and fifth terms of a geometric series are 750 and \(-6\) respectively - Edexcel - A-Level Maths Pure - Question 5 - 2011 - Paper 3
Question 5
The second and fifth terms of a geometric series are 750 and \(-6\) respectively. Find (a) the common ratio of the series, (b) the first term of the series, (c) ... show full transcript
Worked Solution & Example Answer:The second and fifth terms of a geometric series are 750 and \(-6\) respectively - Edexcel - A-Level Maths Pure - Question 5 - 2011 - Paper 3
Step 1
Find the common ratio of the series
Answer
Let the first term of the geometric series be (a) and the common ratio be (r).
From given information: [ ar = 750 \tag{1} ] [ ar^4 = -6 \tag{2} ]
Dividing equation (2) by equation (1): [ \frac{ar^4}{ar} = \frac{-6}{750} ] [ r^3 = \frac{-6}{750} ] [ r^3 = -\frac{1}{125} ] [ r = -\frac{1}{5} ]
Step 2
Step 3
Find the sum to infinity of the series
Answer
The sum to infinity of a geometric series is given by: [ S_\infty = \frac{a}{1 - r} ] Substituting the values of (a) and (r): [ S_\infty = \frac{-3750}{1 - (-\frac{1}{5})} ] [ S_\infty = \frac{-3750}{1 + \frac{1}{5}} = \frac{-3750}{\frac{6}{5}} ] [ S_\infty = -3750 \times \frac{5}{6} = -3125 ]