Question

4. The sum of 50 consecutive positive even integers is 3250. What is the largest of these integers?

Answer 1

114

Explanation:

The question relates to the sum of an arithmetic progression, A. P.

The count of numbers added, n = 50

The sum of the numbers, Sₙ = 3,250

The formula for the sum of an A. P. is given as follows;

`S_n = (n)/(2) \left[2 \cdot a + (n - 1)\cdot d\right]`

`S_n = (n)/(2) \left[a + a_n\right]`

Where;

a = The first term of set of numbers

d = The common difference between consecutive numbers

n = The number of terms = 50

aₙ = The last term, the largest of the integers

Therefore, we get;

The common difference of consecutive even integers numbers, d = 2

Plugging in the values gives;

3,250 = (50/2) × (2·a + (50 - 1) × 2) = 25 × (2·a + 49 × 2)

2·a = (3,250/25) - 49 × 2 = 32

a = 32/2 = 16

From
`S_n = (n)/(2) \left[a + a_n\right]`, we have;

3,250/25 = 16 + aₙ

aₙ = 3,250/25 - 16 = 114

The largest of the integers, aₙ = 114.