Answer:
The number of terms n in arithmetic series is n=25
Step-by-step explanation:
We are given:
a₁ = 13
d = 3
Sₙ = 1225
We need to find number of terms i.e n
The formula used will be: [tex]S_n=\frac{n}{2}(2a+(n-1)d)[/tex]
Putting values and finding n
[tex]S_n=\frac{n}{2}(2a+(n-1)d)\\1225=\frac{n}{2}(2(13)+(n-1)3)\\1225=\frac{n}{2}(26+3n-3)\\1225=\frac{n}{2}(23+3n)\\2(1225)=n(23+3n)\\2450=23n+3n^2\\3n^2+23n-2450=0[/tex]
Now, solving using quadratic formula to find value of n:
[tex]n=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
Putting values: a=3, b=23 and c=-2450
[tex]n=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\n=\frac{-23\pm\sqrt{(23)^2-4(3)(-2450)}}{2(3)}\\n=\frac{-23\pm 173}{6}\\n=\frac{-23+ 173}{6}, n=\frac{-23- 173}{6}\\n=25, n=-\frac{98}{3}[/tex]
So, we get: [tex]n=25, n=-\frac{98}{3}[/tex]
Since number of terms can't be negative so, n= 25
So, the number of terms n in arithmetic series is n=25
