(i) Which term of the sequence 24,2314,2212,2234,…… is the first negative term ?
(ii) Which term of the sequence 12+8i,11+6i,10+4i,…… is (a) purely real (b) purely imaginary ?
The given sequence is 24,2314,2212,2234,……
Here, a = 24
d=2314−24=93−964=−34
Let nth term be the 1st negative term.
an<0
⇒a+(n−1)d<0
⇒24−34(n−1)<0
⇒96−3n+3<0
⇒99<3n
⇒33<n Or n>33
∴ 34th term is 1st negative term.
(ii) (a) 12 + 8i, 11 + 6i, 10 + 4i, ......
This is an A.P.
Here, we have:
a = 12 + 8i
d = (11 + 6i - 12 - 8i)
= (-1 - 2i)
Let the real term be an=a+(n−1)d
an=(12+8i)+(n−1)(−1−2i)
=(12+8i)+(−n+1−2in+2i)
=12+8i−n+1−2in+2i
=(13−n)+(8−2n+2)i
=(13−n)+(10−2n)i
an has to be real.
∴(10−2n)=0
⇒n=5
(b) 12 + 8i, 11 + 6i, 10 + 4i, .....
This is an A.P.
Here, we have:
a = 12 + 8i
d = (11 + 6i - 12 - 8i)
= (-1 -2i)
Let the imaginary term be an=a+(n−1)d
an=(12+8i)+(n−1)(−1−2i)
=(12+8i)+(−n+1−2in+2i)
=12−8i−n+1−2in+2i
=(13−n)+(8−2n+2)i
=(13−n)+(10−2n)i
an has to be imaginary.
∴(13−n)=0
⇒n=13