Theory question 111
Sequences and Series(a) The sum of the first three terms of a decreasing exponential sequence (GP) is equal to 7 and the product of these three terms is equal to 8. Find the: (i) common ratio; (ii) first three terms of the sequence.
(b) Using the trapezium rule with ordinates at x = 1, 2, 3, 4 and 5, calculate, correct to two decimal places, the value of ∫15 (x + 2/x2) dx.
Show the model answer
Model answer
(a) Let the terms be a/r, a, ar. Product a³ = 8, so a = 2.
2/r + 2 + 2r = 7 gives 2r² − 5r + 2 = 0, r = ½ or 2. Decreasing: (i) r = ½.
(ii) 4, 2, 1.
(b) y = 3, 2.5, 3.2222, 4.125, 5.08 at x = 1 to 5; h = 1.
Area ≈ ½(3 + 5.08) + 2.5 + 3.2222 + 4.125 = 13.89.
Explanation
Writing the terms as a/r, a, ar makes the product easy.