MATH SOLVE

2 months ago

Q:
# The 9th term of an arithmetic series is 5/2The sum of the 2nd term and the 5th term of this series is 27Find the sum of the first 100 terms of the series.β

Accepted Solution

A:

Answer:- 8050Step-by-step explanation:The n th term of an arithmetic sequence is[tex]a_{n}[/tex] = a + (n - 1)dwhere a is the first term and d the common difference.We require to find both a and dGiven the 9 th term is 2.5 , thena + 8d = 2.5 β (1)Given the sum of the second and fifth term is 27, thena + d + a + 4d = 27, that is2a + 5d = 27 β (2)Multiply (1) by - 2 and add to (2) to eliminate a- 2a - 16d = - 5 β (3)Add (2) and (3) term by term- 11d = 22 ( divide both sides by - 11 )d = - 2Substitute d = - 2 into (1) and solve for aa - 16 = 2.5 ( add 16 to both sides )a = 18.5The sum to n terms of an arithmetic sequence is[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex][ 2a + (n - 1)d ], thus[tex]S_{100}[/tex] = 50 [ (2 Γ 18.5) + (99 Γ - 2) ] = 50(37 - 198) = 50(- 161) = - 8050