Question

solve the AP: 1+6+11+16+........+x=148 ,find the value of x and n? step by step

Answer 1

Given 1 + 6+ 11+ 16 + ............+x = 148
take the AP :1,6,11,16,..........,x
In this AP
a = 1
d = 6-1 = 5
Given Sn = 148
we know that Sn = n/2[2a+ (n-1)d]
⇒n/2 [ 2a +(n-1)d] = 148
⇒n[2(1) + (n-1)5 ] = 148 ×2
⇒n[ 2 +5n - 5 ] = 296
⇒n [ -3 + 5n ] = 296
⇒ -3n + 5n² = 296
⇒ 5n² - 3n -296 = 0
⇒ 5n² - 40n + 37n - 296 = 0 
⇒5n( n - 8) + 37( n - 8) = 0
⇒ (n - 8) (5n + 37) = 0 
⇒n-8 = 0                or                 5n +37 = 0
⇒n = 8                   or                   n= -37/5
As n is the no.of terms in the AP can not be fractional   and negative 
 ∴n=8
Theirfore x is the 8th term
Tn = T8 = a + (n-1)d 
              = 1 + (8-1)5
              = 1+7(5)
              = 1+35
              = 36
∴8th term = x = 36.