# Lecture-Solution ```Arithmetic Sequence
a, a + d, a + 2d, a + 3d…..
an = a + (n – 1) d
a. Find the nth term of the sequence 2, 5, 8…..
a =2
d=3
an = a + (n – 1) d
an = 2 + (n – 1) 3
b. Find the 25th term in the sequence of 5, 11, 17, 23, 29 . . .
n =25 a = 5 d = 6
a2 – a1 = 11 - 5 = 6
an = a + (n – 1) d
a25 = a + (n – 1) d
a25 = 5 + (25 – 1) 6
a25 = 5 + (24) 6
a25 = 5 + 144
a25 = 149
25th term of the given sequence
c. Find the first term of an arithmetic sequence
if the 9th term is 72 and the common difference is 5.
a9 = 72 n = 9
d =5
a1 =?
an = a + (n – 1) d
72 = a1 + (9 -1)5
72 = a1 + (8)5
72 = a1 + 40
72 – 40 = a1
32 = a1
1st term of the given sequence
a1 = 32
1st term of the given sequence
Geometric Sequence
a, ar, ar2, ar3, ar4….
an = arn-1
r=
𝑎𝑛+1
𝑎𝑛
1. Find the nth term, the fifth term, and the 100th term,
1
of the geometric sequence determined by a = 6 and r = 3 .
a. For nth term
an = arn-1
1
3
an = 6( )n-1
6
an = (3)n-1
an = 2𝑛−1
nth term of the geometric sequence
b. For fifth term
an = arn-1
1
a5 = 6(3)5-1
1
a5 = 6(3)4
1
a5 = 6(34 )
1
a5 = 6(81)
a5 =
a5 =
6
81
2
27
5th term of the geometric sequence
c. For 100th term
an = arn-1
1
a100 = 6(3)100-1
1
a100 = 6(3)99
1
a100 = 6(399 )
6
a100 = 399 =
a100 =
2𝑥3
399
2
398
=
2𝑥3
399
2
= 398
100th term of the geometric sequence
2. Find the common ratio, the fifth term and the nth term
of the geometric sequence -1, 9, -81, 729….
a. For common ratio
a = -1
9
r = −1 = -9 common ratio
b. For fifth term
a5 = a4 ● r
a5= 729 (-9)
a5 = -6,561
5th term of the geometric sequence
c. For nth term
an = arn-1
an = (-1)(-9) n-1
nth term of the geometric sequence
```