in an arithmetic series,the sum of the first 14 terms is -203 and the sum of the next 11 terms is -572.Find the arithmetic sequence
Answers
Answer: It is given that the sum of first 11 terms is 44 that is S
11
=44.
We know that the sum of an arithmetic series with first term a and common difference d is S
n
=
2
n
[2a+(n−1)d], therefore,
S
11
=
2
11
[2a+(11−1)d]
⇒44×2=11(2a+10d)
⇒88=11(2a+10d)
⇒
11
88
=2a+10d
⇒2a+10d=8......(1)
It is also given that the sum of next 11 terms is 55 that is
S
22
=S
11
+55
⇒S
22
=44+55(∵S
11
=44)
⇒S
22
=99
Now the sum of first 22 terms is 99, therefore,
S
22
=
2
22
[2a+(22−1)d]
⇒99×2=22(2a+21d)
⇒198=22(2a+21d)
⇒
22
198
=2a+21d
⇒2a+21d=9......(2)
Subtract equation 1 from equation 2 as follows:
(2a−2a)+(21d−10d)=9−8
⇒11d=1
⇒d=
11
1
Now, substitute the value of d in equation 1:
2a+(10×
11
1
)=8
⇒2a+
11
10
=8
⇒22a+10=88
⇒22a=88−10
⇒22a=78
⇒a=
22
78
⇒a=
11
39
Therefore, the terms of the arithmetic series are:
a
1
=
11
39
a
2
=a
1
+d=
11
39
+
11
1
=
11
40
a
3
=a
2
+d=
11
40
+
11
1
=
11
41
Hence, the required arithmetic series is
11
39
+
11
40
+
11
41
+.
Step-by-step explanation: