What will be the second term of an AP if the sum of its first n terms is 2n2 +6n
Answers
Let . And let the n-th term be .
The relation between and is as follows.
- for n≥2
We want , which is the second term.
Using the relation,
Hence, the answer is 12.
More information:
About for two cases
- (without a constant)
This satisfies
over every natural numbers.
To prove this, you can simplify LHS. Then show they always equal by .
- (with a constant)
This satisfies
for n≥2.
To prove this, you can simplify LHS. Then show they never equal for n=1 by .
Answer:
Let S_n=2n^2+6nS
n
=2n
2
+6n . And let the n-th term be a_na
n
.
The relation between S_nS
n
and a_na
n
is as follows.
S_1=a_1S
1
=a
1
S_n-S_{n-1}=a_nS
n
−S
n−1
=a
n
for n≥2
We want a_2a
2
, which is the second term.
Using the relation,
S_2-S_1=a_2S
2
−S
1
=a
2
\Longleftrightarrow 20-8=a_2⟺20−8=a
2
\Longleftrightarrow 12=a_2⟺12=a
2
Hence, the answer is 12.
More information:
About S_nS
n
for two cases
S_n=an^2+bnS
n
=an
2
+bn (without a constant)
This satisfies
S_n-S_{n-1}=a_nS
n
−S
n−1
=a
n
over every natural numbers.
To prove this, you can simplify LHS. Then show they always equal by S_1=a_1S
1
=a
1
.
S_n=an^2+bn+cS
n
=an
2
+bn+c (with a constant)
This satisfies
S_n-S_{n-1}=a_nS
n
−S
n−1
=a
n
for n≥2.
To prove this, you can simplify LHS. Then show they never equal for n=1 by S_1=a_1S
1
=a
1