To find the value of [tex]\( n \)[/tex] for which the sum of the first [tex]\( n \)[/tex] terms ([tex]\( S_n \)[/tex]) of an arithmetic progression (AP) is 116, given the first term ([tex]\( a \)[/tex]) is 25 and the common difference ([tex]\( d \)[/tex]) is -3, we can use the formula for the sum of the first [tex]\( n \)[/tex] terms of an AP:
[tex]\[ S_n = \frac{n}{2} \left( 2a + (n - 1)d \right) \][/tex]
Given:
[tex]\[ a = 25 \][/tex]
[tex]\[ d = -3 \][/tex]
[tex]\[ S_n = 116 \][/tex]
Substituting these values into the sum formula:
[tex]\[ 116 = \frac{n}{2} \left( 2 \cdot 25 + (n - 1) \cdot (-3) \right) \][/tex]
Simplifying within the parentheses:
[tex]\[ 116 = \frac{n}{2} \left( 50 - 3n + 3 \right) \][/tex]
[tex]\[ 116 = \frac{n}{2} \left( 53 - 3n \right) \][/tex]
Multiplying both sides by 2 to clear the fraction:
[tex]\[ 232 = n \left( 53 - 3n \right) \][/tex]
This simplifies to a quadratic equation:
[tex]\[ 232 = 53n - 3n^2 \][/tex]
[tex]\[ 3n^2 - 53n + 232 = 0 \][/tex]
We now have a quadratic equation in standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], with [tex]\( a = 3 \)[/tex], [tex]\( b = -53 \)[/tex], and [tex]\( c = 232 \)[/tex].
We solve this quadratic equation to find the values of [tex]\( n \)[/tex]:
The quadratic formula is:
[tex]\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substituting the values [tex]\( a = 3 \)[/tex], [tex]\( b = -53 \)[/tex], and [tex]\( c = 232 \)[/tex]:
[tex]\[ n = \frac{-(-53) \pm \sqrt{(-53)^2 - 4 \cdot 3 \cdot 232}}{2 \cdot 3} \][/tex]
[tex]\[ n = \frac{53 \pm \sqrt{2809 - 2784}}{6} \][/tex]
[tex]\[ n = \frac{53 \pm \sqrt{25}}{6} \][/tex]
[tex]\[ n = \frac{53 \pm 5}{6} \][/tex]
This gives us two solutions:
[tex]\[ n = \frac{53 + 5}{6} = \frac{58}{6} = \frac{29}{3} \][/tex]
[tex]\[ n = \frac{53 - 5}{6} = \frac{48}{6} = 8 \][/tex]
So, the values of [tex]\( n \)[/tex] are:
[tex]\[ n = 8 \quad \text{or} \quad n = \frac{29}{3} \][/tex]
However, [tex]\( n \)[/tex] must be a positive integer because it represents the number of terms. Therefore, the valid solution is:
[tex]\[ n = 8 \][/tex]
