Question

if Sn the sum of first n terms of an AP is given by Sn =(3n2-4n ) then find its 25th term​

Answer 1

Given :-

◉ The sum of first n terms of an AP is given by Sₙ = 3n² - 4n

To Find :-

◉ 25th term of the AP

Solution :-

For finding the 25th term, First we should find the common difference and first term of the AP.

For the first term, put n = 1 in the given formula,

⇒ S₁ = 3(1)² - 4(1)

⇒ S₁ = 3 - 4

⇒ S₁ = -1

But, It is the first term hence there would be nothing added to it:

∴ a₁ = -1 ...(1)

Similarly, Let us find the sum of first two terms, put n = 2,

⇒ S₂ = 3(2)² - 4(2)

⇒ S₂ = 12 - 8

⇒ S₂ = 4 ...(2)

We needed the second term but what we actually got is the sum of first two terms of the AP, So subtract the first term from it to get the second term:

⇒ a₂ = 4 - a₁

⇒ a₂ = 4 - (-1)

⇒ a₂ = 4 + 1

⇒ a₂ = 5 ...(3)

Now, Let us find the common difference, But It is given that It is an AP that we are asked to find the 25th term of, So we don't need to find the Difference of third and second term, Instead difference of first and second term would be fine because in an AP the difference between any two consecutive terms is same.

⇒ d = a₂ - a₁

⇒ d = 5 - (-1) [ from (1) & (3) ]

⇒ d = 5 + 1

⇒ d = 6

Now let us find the 25th term,

⇒ aₙ = a + (n - 1)d

⇒ a₂₅ = a + (25 - 1)d

⇒ a₂₅ = a + 24d

Substituting values,

⇒ a₂₅ = -1 + 24×6

⇒ a₂₅ = -1 + 144

⇒ a₂₅ = 143

Hence, The 25th term of the AP is 143.