S_n=n/2[2a+(n-1)d]
where a is the first term, d is the difference between 2 adjacent terms and n is the nth term
Looking at the sequence: 2.5, 4, 5.5, ...
a=2.5
d=4-2.5=1.5
Putting those two constants into the equation:
S_n=n/2[2(2.5)+(n-1)(1.5)]
S_n=n/2[5+1.5n-1.5]
S_n=n/2[1.5n+3.5]
Now to find the number of terms needed for the sum to be greater than 200,
ie S_n > 200
n/2[1.5n+3.5] >200
(n is the unknown variable that we have to find)
n[1.5n+3.5] >200times2
1.5n^2+3.5n >400
1.5n^2+3.5n-400 >0
0.3n^2+0.7n-80>0
Using the quadratic formula,
n=(-0.7+-sqrt(0.7^2-4(0.3)(-80)))/(2times0.3)
n=(-0.7+-sqrt(0.49+96))/0.6
n=(-0.7+-sqrt96.49)/0.6
n=(-0.7+sqrt96.49)/0.6 only as n>0 since n is the number of terms
n=15.20488725
That means that we must take a minimum of 16 terms in order for our sum to be greater than 200
To test, we sub n=15 and n=16 back into our equation S_n=n/2[1.5n+3.5]
If n=15,
S_15=15/2[1.5times15+3.5]
S_15=195
If n=16,
S_16=16/2[1.5times16+3.5]
S_16=220
Therefore, 16 terms are required is correct