Answer:
[tex]n =16[/tex] --- Number of terms
[tex]d = \frac{8}{3}[/tex] --- Common difference
Explanation:
Given
[tex]a = 5[/tex] --- first term
[tex]T_n = 45[/tex]
[tex]S_n = 400[/tex]
Required
Determine the number of terms (n) and the common difference (d)
The sum of n terms of an AP is:
[tex]S_n = \frac{n}{2}(a + T_n)[/tex]
This gives:
[tex]400 = \frac{n}{2}(5 + 45)[/tex]
[tex]400 = \frac{n}{2} * 50[/tex]
[tex]400 = n * 25[/tex]
Divide both sides by 25
[tex]16 = n[/tex]
[tex]n =16[/tex]
The nth term of an AP is:
[tex]T_n = a + (n - 1)d[/tex]
This gives:
[tex]45 = 5 + (16 - 1) * d[/tex]
[tex]45 = 5 + 15 * d[/tex]
Subtract 5 from both sides
[tex]45 - 5= 5 - 5 + 15 * d[/tex]
[tex]40= 15 * d[/tex]
Divide both sides by 15
[tex]\frac{40}{15} = d[/tex]
[tex]\frac{8}{3} = d[/tex]
[tex]d = \frac{8}{3}[/tex]
