Math  /  Numbers & Operations

Questionk=400500k\sum_{k=400}^{500} k. (Hint: the sum of inte the sum of the first 50 integer

Studdy Solution

STEP 1

1. We are summing consecutive integers from k=400 k = 400 to k=500 k = 500 .
2. The formula for the sum of an arithmetic series can be used.
3. The series is arithmetic with a common difference of 1 1 .

STEP 2

1. Identify the arithmetic series parameters.
2. Use the arithmetic series sum formula.
3. Calculate the sum.

STEP 3

Identify the first term a a and the last term l l of the series.
- First term a=400 a = 400 - Last term l=500 l = 500

STEP 4

Determine the number of terms n n in the series.
The number of terms n n is given by:
n=la+1=500400+1=101 n = l - a + 1 = 500 - 400 + 1 = 101

STEP 5

Use the formula for the sum of an arithmetic series:
Sn=n2×(a+l) S_n = \frac{n}{2} \times (a + l)
Substitute the known values:
S101=1012×(400+500) S_{101} = \frac{101}{2} \times (400 + 500)

STEP 6

Calculate the sum:
S101=1012×900=101×450=45450 S_{101} = \frac{101}{2} \times 900 = 101 \times 450 = 45450
The sum of the integers from 400 400 to 500 500 is:
45450\boxed{45450}

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