#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
