In an A.P, the sum of the first n terms is 3n^2-2n.find its A.P
Answers
Answer:
A.P is 1, 7, 13, 19....
Explanation:
To find the Arithmetic progression, the two values required are the common difference 'd' and and the first term 'a'. To get it, we substitute 'n' with 1, 2, and so on to derive values for 'd' and 'a' later.
Before answering this question, make sure you've understood this.
→ The First term of an AP is equal to S₁
→ The Second term of an AP is equal to S₂ - S₁
→ The Third term of an AP is equal to S₃ - S₂
→ And so on.
Given that,
Sn = 3n² - 2n.
Case I:
Let us take n = 1.
Sn = 3n² - 2n.
S₁ = 3(1)² - 2(1)
S₁ = 3 - 2
S₁ = 1
Case II:
Let us take n = 2.
Sn = 3n² - 2n.
S₂ = 3(2)² - 2(2)
S₂ = 3(4) - 4
S₂ = 12 - 4
S₂ = 8
Case III:
Let us take n = 3.
Sn = 3n² - 2n.
S₂ = 3(3)² - 2(3)
S₂ = 3(9) - 6
S₂ = 27 - 6
S₂ = 21
As we stated in the beginning, the first term is equal to the sum of the first terms. And, the second term is equal to the Sum of terms upto 2 subtracted by Sum of terms upto 1.
a₁ = S₁
a₁ = 1
a₂ = S₂ - S₁
a₂ = 8 - 1
a₂ = 7
With the first two terms, we can find the common difference.
d = a₂ - a₁
d = 7 - 1
d = 6
Now that we have the first term and the common difference, the AP will be:
A.P → 1, 7, 13, 19.........