Home
Class 12
MATHS
if (m+1)th, (n+1)th and (r+1)th term of ...

if `(m+1)th, (n+1)th and (r+1)th` term of an AP are in GP.and m, n and r in HP. . find the ratio of first term of A.P to its common difference

Text Solution

Verified by Experts

Let a be the first term and d be the common difference of the given A.P. Then, as given the (m+1)th,(n+1)th, and (r+1)th terms are in G.P. So,
a+md,a+nd,a+rd are in G.P.
`rArr(a+nd)^(2)=(a+md)(a+rd)`
`rArra(2n-m-r)=d(mr-n^(2))`
`rArrd/a=(2n-(m+r))/(mr-n^(2))` (1)
Next, m ,n r are in H.P. Hence,
`n=(2mr)/(m+r)` (2)
From (1) and (2),
`d/a(2n-(m+r))/(mr-n^(2))=2/n((2n-(m+r))/((m+r)-2n))=-2/n`
Promotional Banner

Topper's Solved these Questions

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise ILLUSTRATION 5.72|1 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise ILLUSTRATION 5.73|1 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise ILLUSTRATION 5.70|1 Videos

  • PROBABILITY II

    CENGAGE PUBLICATION|Exercise MULTIPLE CORRECT ANSWER TYPE|6 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Archives (Numerical Value Type)|3 Videos

Similar Questions

Explore conceptually related problems

If the (m + 1)th, (n + 1)th and (r + 1)th terms of an A.P. are in G.P. and (1)/(m), (1)/(n), (1)/(r ) are in A.P., then find the ratio of the first term of the A.P. to its common difference in terms of n.

If (m+1)^th , (n+1)^th and (r+1)^th terms in AP are in GP m,n,r are in HP ,show that the ratio of the common difference to the first term of the AP is (-2/n).

The sum of n terms of an A.P is n^2 .Find the common difference.

The 5th and 13th terms of an A.P. are 16 and 28 respectively. Find the A.P.

The 10th term of an A.P. is (-15) and the 31st term is (-57). Find the first term and common difference of the A.P.

The 5th and 13th term of an A.P are 5 and -3 respectively. Find the A.P and the 30th term.

The first term of G.P. is 27 and 8th term is 1/81 . Find the sum of its first 10 terms.

If the 2nd ,5ht and 9th terms of a non-constant AP are in GP , then the common ratio of this GP is

The sum to n terms of an A.P. is n^(2) . Find the common difference.

The sums of n terms of two A.P.'s are in the ratio (4n-13) : (3n + 10). Find the ratio of their 9th terms.