Mathematics 2014 JAMB Past Questions - Question 20: what is the common ratio of G.P. ( √10 +√5)+ (√10+2√5)+....?

Correct Answer

B

Explanation

To find the common ratio of the geometric progression (G.P.) (√10 + √5) + (√10 + 2√5) + ..., we need to examine the pattern of the terms.Let's observe the terms of the G.P.:First term: (√10 + √5)Second term: (√10 + 2√5)Third term: (√10 + 3√5)...We can see that each term is obtained by adding an increasing multiple of √5 to (√10). Therefore, the common ratio (r) can be determined by dividing the second term by the first term:r = (√10 + 2√5) / (√10 + √5)To simplify this expression, we can multiply the numerator and denominator by the conjugate of the denominator, which is (√10 - √5):r = [(√10 + 2√5) / (√10 + √5)] * [(√10 - √5) / (√10 - √5)]Expanding the numerator and denominator, we get:r = [(√10 * √10) + (√10 * -√5) + (2√5 * √10) + (2√5 * -√5)] / [(√10 * √10) + (√10 * -√5) + (√10 * √5) + (√5 * -√5)]Simplifying further, we have:r = [10 - 5√2 + 2√50 - 10] / [10 - √50 + √50 - 5]The terms with √2 and √50 cancel out, and the terms with 10 and -10 also cancel out:r = [-5√2 + 2√50] / [-5]Finally, we can simplify this expression by dividing both the numerator and denominator by 5:r = (√2 - 2√10) / 1Therefore, the common ratio of the given geometric progression is (√2 - 2√10).