If the third and ninth terms of an A.P. are 1 and 19 respectively, th…

The correct answer is Option (2) → 61

The \(n\)th term of an arithmetic progression is \(a_n=a+(n-1)d\), where \(a\) is the first term and \(d\) the common difference. The two given terms give two equations, and subtracting them removes \(a\).

The two equations (from the given terms):

• Third term: \(a+2d=1\)        (1)

• Ninth term: \(a+8d=19\)        (2)

Finding \(d\) and \(a\) (subtracting (1) from (2), the \(a\) terms cancel):

• \(6d=18\), so \(d=3\)

• Putting \(d=3\) back into (1): \(a=1-2\times3=-5\)

The 23rd term:

• \(a+22d=-5+22\times3=61\)