Question

between 1 and 31 are inserted m by arthematic mean so that the ratio of 7th and m-1th term is 5:9 find m

Answer 1

Sol:
1) Terms of the A.P. are 1, A1, A2, A3 ....Am , 31

a = 1 , an = 31 and n = m+2
 
an = a + (n -1)d
 
31 = 1 + (m + 2 - 1)d
 
30 = (m + 1)d

d = 30/(m + 1)
 
A7 = a + 7d = 1 + 7[30/(m+1)]  --------------------1
 
Am-1 = a + (m-1)d = 1 + (m-1)[30/(m+1)]  ---------2
 
From 1 and 2,
 
A7/Am-1
 
⇒ 1 +7[30/(m+1) / 1+(m-1)[30/(m+1)] = 5/9
 
⇒ [m + 1 + 7(30)] / [m + 1 + 30 m – 30] = 5/9
 
 ⇒ [m + 211] / [31 m – 29] = 5/9
 
⇒ 9 m + 1899 = 155 m – 145
 
⇒ 146 m = 2044
 
⇒ m = 14
 
Hence, the value of m is 14.