Answer : (d) 59
Let the first term, common difference and number of terms of the A.P. be a, d and n respectively.
Then,
a + d = \(7\frac{3}{4}\) ... (i)
a + 30d = \(\frac{1}{2}\) ...(ii)
and a + (n – 1)d = − \(6\frac{1}{2}\) ...(iii)
Eqn (ii) – Eqn (i)
⇒ 29d = \(\frac{1}{2}\) - \(\frac{31}{4}\) = - \(\frac{29}{4}\)
⇒ d = - \(\frac{1}{4}\)
Putting d = - \(\frac{1}{4}\) in (i), we get
a - \(\frac{1}{4}\) = \(7\frac{3}{4}\) + \(\frac{1}{4}\)
⇒ a = 8
∴ Putting the values of a and d in (iii), we have
8 + (n – 1) ( - \(\frac{1}{4}\) ) = - \(\frac{13}{2}\)
⇒ 8 + \(\frac{1}{4}\)- \(\frac{1}{4}\)n = - \(\frac{13}{2}\)
⇒ −\(\frac{1}{4}\)n = − \(\frac{13}{2}\) − \(\frac{33}{4}\)
⇒ − \(\frac{1}{4}\)n = − \(\frac{59}{4}\)
⇒ n = 59
