Determining Common Difference, Given the Sum of an Arithmetic Series

💎 Using particular sum formulae as follows:
(i). Snn(a+)
(ii). Snn(a+an)
(iii). Snn(2a+(n-1)d)
(iv). Sn=Sn-1+an, i.e., an=SnSn-1

📌 Example 1. In an arithmetic sequence, the first term is 2, the last term is 29 and the sum of all the terms is 155. Find the common difference.
✍ Solution:
Here, a=2, =29 and Sn=155.
Let d be the common difference of the given arithmetic sequence and n be the total number of terms. Then, an=29.

a+(n-1)d=29
2+(n-1)d=29 … (i)

The sum of n terms of an arithmetic sequence is given by
Snn(a+)=155
½n(2+29)=½⋅n⋅31=155
n=10.

Putting the value of n in (i), we get
=2+9d=29
9d=27
d=3.

Thus, the common difference of the given arithmetic sequence is 3.

📌 Ex2. In an arithmetic sequence, the first term is -4, the last term is 29 and the sum of all its terms is 150. Find its common difference.
✍ Solution:
Suppose there are n terms in the arithmetic sequence.
Here, a=-4, =29 and Sn=150.

Snn(a+)
½n(-4+29)=150
n=(150⋅2)/25=12.

Thus, the arithmetic sequence contains 12 terms. Let d be the common difference of the arithmetic sequence.
a12=29
[an=a+(n-1)d]
-4+(12-1)⋅d=29
11d=29+4=33
d=3.

Hence, the common difference of the arithmetic sequence is 3.

📌 Ex3. The first and last terms of an arithmetic sequence are 5 and 45 respectively. If the sum of all its terms is 400, find the common difference and the number of terms.
✍ Solution:
Suppose there are n term in the arithmetic sequence.
Here, a=5, =45 and Sn=400.

Sn=400
Snn(a+)
½n(5+45)=400
n⋅½⋅50=400
n=400÷25
n=16.

Thus, there are 16 terms in the arithmetic sequence. Let d be the common difference of the arithmetic sequence.
a16=45
[an=a+(n-1)d]

5+(16-1)⋅d=45
15d=45-5=40
d=40/15=8/3.

Hence, the common difference of the arithmetic sequence is 8/3.

📌 Ex4. In an arithmetic sequence: given a=8, an=62, Sn=210, find n and d.
✍ Solution:
Here, a=8, an=62 and Sn=210.
The sum of n terms of an arithmetic sequence is given by

Snn(a+an)
210=½n⋅(8+62)
210=35n
6=n
an=a+(n-1)d
62=8+(6-1)d
54=5dd=54/5

📌 Ex5. In an arithmetic sequence: given a=3, n=8, S=192, find d.
✍ Solution:
Here, a=3, n=8 and S=192. The sum of n terms of an arithmetic sequence is given by

Snn(a+an)
192 =8/2[3+an]
192=4[3+an]
3+an=192/4=48
an=45
an=a+(n-1)d
45=3+(8-1)d
42=7d
d=6

📌 Ex6. An arithmetic progression has 3 as its first term. Also, the sum of the first 8 terms is twice the sum of the first 5 terms. Find the common difference.
✍ Solution:
We are given that a=3. We are also given some information about the sums S8 and S5, and we want to find the common difference. So we shall use the formula

Snn(2a+(n-1)d)

for the sum of the first n terms. This tells us that
S8=½×8×(6+7d)

and that
S5=½×5×(6+4d).

So, using the given fact that S8=2S5, we see that
½×8×(6+7d)=2×½×5×(6+4d)
4×(6+7d)=5×(6+4d)
24+28d=30+20d
8d=6
d

📌 Ex7. In an arithmetic sequence, the first term is 22, nth terms is -11 and sum of first n terms is 66. Find the number n and the common difference.
✍ Solution:
Here, a=22, an=-11 and Sn=66.
Let d be the common difference of the given arithmetic sequence.
Then, an=-11

a+(n-1)d=-11
=22+(n-1)d=-11
(n-1)d=-33 … (i)

The sum of n terms of an arithmetic sequence is given by
Snn[2a+(n-1)d]=66
[Substituting the value off (n-1)d from (i)]
½n[2⋅22+(-33)]=½n⋅11=66
n=12.

Putting the value of n in (i), we get:
11d=-33
d=-3.

Thus, n=12 and d=-3.

📌 Ex8. If the sum of first n terms is (3n2+5n), find its common difference.
✍ Solution:
Let Sn denotes the sum of first n terms of the arithmetic sequence.

Sn=3n2+5n
Sn-1=3(n-1)2+5(n-1)
=3(n2-2n+1)+5(n-1)
=3n2n-2.

Now, nth term of arithmetic sequence, an=SnSn-1
=(3n2+5n)-(3n2n-2)
=6n+2.

Let d be the common difference of the arithmetic sequence.
d=ana(n-1)
=(6n+2)-[6(n-1)+2]
=6n+2-6(n-1)-2
=6.

Hence, the common difference of the arithmetic sequence is 6.

📌 Ex9. The sum of first n terms of an arithmetic sequence is (3n2+6n). The common difference of the arithmetic sequence is

(a) 6 (b) 9 (c) 15 (d) -3

✍ Solution:
Let Sn denotes the sum of first n terms of the arithmetic sequence.
Sn=3n2+6n
Sn-1=3(n-1)2+6(n-1)
=3(n2-2n+1)+6(n-1)
=3n2-3.

So, nth term of the arithmetic sequence, an=SnSn-1=(3n2+6n)-(3n2-3).
=6n+3.
Let d be the common difference of the arithmetic sequence.
d=ana(n-1)=(6n+3)-[6(n-1)+3]
=6n+3-6(n-1)-3
=6.

Thus, the common difference of the arithmetic sequence is 6.
Answer: (a) 6

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