Let the first term, common difference and number of terms of the AP are a, d and n, respectively.
We know that, if last term of an AP is known, then,
l = a + (n - 1)d . . . . . (i)
And nth term of an AP is,
Tn = a + (n - 1)d . . . . . (ii)
Given that, 26th term of an AP = 0
⇒ T26=a+(26−1)d=0 [from eq.(i)]
⇒ a + 25d = 0 . . . . (iii)
11th term of an AP = 3
⇒ T11=a+(11−1)d=3 [from eq.(ii)]
⇒ a + 10d = 3 . . . . .(iv)
Last term of the AP, l=−15
⇒ l = a + (n - 1)d [From eq. (i)]
⇒ +15 = a + (n - 1)d . . . . . (v)
Now, subtracting eq.(iv) from eq.(iii),
a+25d=0a+10d=3− − − –––––––––––––15d=−3
⇒ d=−15
Put the value of d in eq.(iii), we get;
a+25(−15)=0⇒ a−5=0⇒a=5
Now, put the value of a, d in eq.(v), we get;
−15=5+(n−1)(−15)
⇒ -1 = 25 - (n - 1)
⇒ -1 = 25 - n + 1
⇒ n = 25 + 2 = 27
Hence, the common difference and number of term are −15 and 27, respectively.