Advertisements
Advertisements
Question
If a, b, c are in G.P. and a, x, b, y, c are in A.P., prove that `a/x + c/y = 2`
Advertisements
Solution
a, b and c are in G.P.
`=>` b2 = ac
a, x, b, y and c are in A.P.
`=>` 2x = a + b `=> x = (a + b)/2`
2b = x + y `=> b = (x + y)/2`
2y = b + c `=> y = (b + c)/2`
Now,
`a/x + c/y = (2a)/(a + b) + (2c)/(b + c)`
= `(2a(b + c) + 2c(a + b))/((a + b)(b + c))`
= `(2ab + 2ac + 2ac + 2bc)/(ab + ac + b^2 + bc)`
= `(2ab + 4ac + 2bc)/(ab + b^2 + b^2 + bc)`
= `(2(ab + 2ac + bc))/(ab + 2b^2 + bc)`
= `(2(ab + 2ac + bc))/(ab + 2ac + bc)`
= 2
RELATED QUESTIONS
If the first and the third terms of a G.P. are 2 and 8 respectively, find its second term.
The fifth, eight and eleventh terms of a geometric progression are p, q and r respectively. Show that : q2 = pr.
For the G.P. `1/27, 1/9, 1/3, ........., 81`; find the product of fourth term from the beginning and the fourth term from the end.
Q 8
If each term of a G.P. is raised to the power x, show that the resulting sequence is also a G.P.
Q 2
Q 7
The first two terms of a G.P. are 125 and 25 respectively. Find the 5th and the 6th terms of the G.P.
Find the sum of the sequence `-1/3, 1, -3, 9, ..........` upto 8 terms.
Find a G.P. for which the sum of first two terms is – 4 and the fifth term is 4 times the third term.
