How many terns of the arithmetic sequence 2.5,4,5.5... must be taken for the sum to be greater than 200?

1 Answer
Jul 27, 2018

16 terms are required

Explanation:

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