The sum of first 15th term of an arithmetic sequence is 570 and its 12th trem is 62
What is the sum of 30th term
Answers
We know that
↝ Sum of n terms of an arithmetic sequence is,
Wʜᴇʀᴇ,
Sₙ is the sum of n terms of AP.
a is the first term of the sequence.
n is the no. of terms.
d is the common difference.
Now, Given that
Sum of first 15 terms of an AP = 570
Also, we know that
↝ nᵗʰ term of an arithmetic sequence is,
Wʜᴇʀᴇ,
aₙ is the nᵗʰ term.
a is the first term of the sequence.
n is the no. of terms.
d is the common difference.
Now, Given that
On Subtracting equation (1) from equation (2), we get
On substituting the value of d, in equation (1) we get
Now, Sum of first 30 terms is given by
Answer:
We know that
s
n
=
n
2
(
2
a
+
(
n
−
1
)
d
)
Thus
186
=
12
2
(
2
a
+
(
11
)
d
)
186
=
6
(
2
a
+
11
d
)
31
=
2
a
+
11
d
Now we know that
t
n
=
a
+
(
n
−
1
)
d
.
83
=
a
+
(
20
−
1
)
d
83
=
a
+
19
d
We now have a system of equations:
{
31
=
2
a
+
11
d
83
=
a
+
19
d
Substituting (2) into (1), we get
31
=
2
(
83
−
19
d
)
+
11
d
31
=
166
−
38
d
+
11
d
−
135
=
−
27
d
d
=
5
Now solving for
a
:
83
−
19
(
5
)
=
−
12
The sum is once again given by
s
40
=
40
2
(
2
(
−
12
)
+
(
39
)
5
)
s
40
=
20
(
171
)
s
40
=
3420
Hopefully this helps!