Answer:
S n = n/2 (2 a 1 +d(n-1) )
n=10
Step-by-step explanation:
4,7,10,.... is an infinite series and Sn for an infinite series is infinity when d>0 or negative infinity when d<0
The formula for the sum of an arithmetic series is given by
S n = n/2 ( a 1 + a n )
and an is found by
an =a1+d(n-1)
Substituting in
S n = n/2 ( a 1 +a1+d(n-1) )
Simplifying
S n = n/2 (2 a 1 +d(n-1) )
Taking the series
4,7,10,.....
a1 =4
7-4 =3
10-7 =3
So d = 3
Sn = 175
175 = n/2 (2*4+3(n-1))
Multiply by 2
175*2 = 2*n/2 (2*4+3(n-1))
350 = n (2*4+3(n-1))
Distribute the 3 and simplify
350 = n(8+3n-3)
350 = n(5+3n)
Distribute the n
350 =5n +3n^2
Subtract 350 from each side
350-350 = 3n^2 +5n -350
0 = 3n^2 +5n -350
Factor
0 = (n - 10) (3 n + 35)
Using the zero product property
n-10 = 0 3n+35=0
n=10 3n=-35
n=-35/3
n cannot be negative
