Question

Between 1 and 31, m number have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m-1)th no. is 5:9 find the value of m.​

Answer 1

Answer:

Since m numbers are inserted between 1 and 31 then the total number of terms in the A.P. = n = (m+2).

The first term = a = 1

The last term =

= a+(m+2-1)d= 31

or, 1+(m+1)d=31

or, (m+1)d=30

or, d=30/(m+1) ---------------------(1)

Now, according to the question,

[a+7d]/[a+(m-1)d]=5/9

or, (1+7d)/{1+(m-1)d}=5/9

or, 9+63d=5+5(m-1)d

or, 63d-5(m-1)d=5-9

or,(63-5m+5)d=-4

or, (68-5m){30/(m+1)}=-4

or, 30(68-5m)=-4(m+1)

or, 2040-150m=-4m-4

or, -150m+4m=-2040-4

or, -146m=-2044

or, m=(-2044)/(-146)

or, m=14 Ans.