if the sum of first n terms of a sequence is 1/4 [n^2(n+1)^2] find the fifth term.?

=1/4 [n^2(n+1)^2]

Let n=5

=1/4 [5^2(5+1)^2]

=1/4 (25*36)

=1/4 * 900

=225

Let n=4

=1/4 [4^2(4+1)^2]

=1/4 (16*25)

=1/4 * 400

=100

5th Term = Sum of first 5 - Sum of first 4

=225-100

=125

Hope this helps!

i) nth term of any sequence is given by: (Sum to n terms) - {Sum to (n-1) terms}

ii) Since you need 5th term, it is: (1/4)[{5²(5+1)²} - {4²(4+1)²}] = (5²/4)(6² - 4²) = 125

iii) However of the above, this problem can be solved in another easy method.

The given sum [(1/4)*{(n²)*(n+1)²}] is the sum of cubes of first n natural numbers.

That is: 1³ + 2³ + 3³ + ............ + n³ = [(1/4)*{(n²)*(n+1)²}]

As such 5th term means n = 5, that is 5³ = 125

Solve for n=4

Solve for n=5

Find the difference.

(.25)(16)(25)=100

(.25)(25)(36)=225

225-100 = 125

The fifth term is (sum of first 5 terms) - (sum of first 4 terms), so put 5 and 4 into the formula and find the differrence.