If for an AP a=25 d=-3a and SN=105 then n=?

Question

Answer

Calculation of n for the given AP

Let's consider the given AP:

Finding the Sum of First n Terms of the AP

The sum of the first n terms of an AP can be calculated using the formula:

Sn = n/2 * [2a + (n-1)d]

So, substituting the given values in the formula, we get:

105 = n/2 * [2(25) + (n-1)(-3a)]

105 = n/2 * [50 - 75(n-1)]

105 = n/2 * [-25n + 125]

210 = -25n^2 + 125n

25n^2 - 125n + 210 = 0

Finding the Value of n

Using the quadratic formula, we can find the value of n:

n = [-b ± √(b^2 - 4ac)]/2a

Substituting the values of a, b, and c in the formula, we get:

n = [125 ± √(125^2 - 4(25)(210))]/2(25)

n = [125 ± √(15625 - 21000)]/50

n = [125 ± √(5625)]/50

n = [125 ± 75]/50

So, n = 5 or n = 2.5. But since n has to be a positive integer, n = 5.

Conclusion

Therefore, the value of n for the given AP is 5.

Answer

Sn=a+(n-1)d
105=25+3n-3
105=3n+22
105-22=3n
83=3n
n=83/3
n=27