Sure! Let's solve the problem step-by-step.
Given:
- First term of the arithmetic progression (A.P), [tex]\( a = 3 \)[/tex]
- Last term of the A.P, [tex]\( l = 90 \)[/tex]
- Sum of the series, [tex]\( S = 1395 \)[/tex]
We need to find the number of terms [tex]\( n \)[/tex] in the A.P.
The sum of an arithmetic progression is given by the formula:
[tex]\[ S = \frac{n}{2} \times (a + l) \][/tex]
Here:
- [tex]\( S \)[/tex] is the sum of the series
- [tex]\( n \)[/tex] is the number of terms in the series
- [tex]\( a \)[/tex] is the first term of the series
- [tex]\( l \)[/tex] is the last term of the series
Substituting the given values into the formula:
[tex]\[ 1395 = \frac{n}{2} \times (3 + 90) \][/tex]
Simplify inside the parentheses:
[tex]\[ 3 + 90 = 93 \][/tex]
So our equation becomes:
[tex]\[ 1395 = \frac{n}{2} \times 93 \][/tex]
To isolate [tex]\( n \)[/tex], first multiply both sides by 2:
[tex]\[ 2 \times 1395 = n \times 93 \][/tex]
[tex]\[ 2790 = n \times 93 \][/tex]
Now, divide both sides by 93 to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{2790}{93} \][/tex]
[tex]\[ n = 30 \][/tex]
Therefore, the number of terms [tex]\( n \)[/tex] in the arithmetic progression is:
[tex]\[ \boxed{30} \][/tex]
