Question

What is the difference between the sums of all even and odd numbers in numbers from 50 to 151?​

Answer 1

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We can also use S = n[2a + (n-1)d]/2 to calculate both answers

where n = number of terms, a = first term, and d = difference between terms.

There are 101 terms/numbers between 50 and 150,

and we can see the start and end numbers of 50 and 150 respectively are even,

therefore there must be 1 more positive number than negative number.

That makes 51 positive numbers and 50 negative numbers from 50 to 150.

So the sum of positive numbers from 50 to 150 is

S = n[2a + (n-1)d]/2

S= 51[2(50) + (50)2]/2

S=51(200)/2

S=5100

Likewise for the negatives,

S = n[2a + (n-1)d]/2

S= 50[2(51) + (49)2]/2

S=50(200)/2

S=5000

To check our work we can use the above formula to calculate the sum of the numbers from 1 to 150,

S = 150[2(1) + 149(1)]/2

S= 150(151)/2 = 11325

and subtract the sum of the numbers from 1 to 49,

S = 49[2(1) + 48(1)]/2

S= 49(50)/2 = 1225 (same as Gauss’ formula of n(n+1)/2)

So the sum of numbers from 50 to 150 = 11325 - 1225 = 10100 (5100 positive + 5000 negative)

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Answer 2

Answer:

We can also use S = n[2a + (n-1)d]/2 to calculate both answers

where n = number of terms, a = first term, and d = difference between terms.

There are 101 terms/numbers between 50 and 150,

and we can see the start and end numbers of 50 and 150 respectively are even,

therefore there must be 1 more positive number than negative number.

That makes 51 positive numbers and 50 negative numbers from 50 to 150.

So the sum of positive numbers from 50 to 150 is

S = n[2a + (n-1)d]/2

S= 51[2(50) + (50)2]/2

S=51(200)/2

S=5100

Likewise for the negatives,

S = n[2a + (n-1)d]/2

S= 50[2(51) + (49)2]/2

S=50(200)/2

S=5000

To check our work we can use the above formula to calculate the sum of the numbers from 1 to 150,

S = 150[2(1) + 149(1)]/2

S= 150(151)/2 = 11325

and subtract the sum of the numbers from 1 to 49,

S = 49[2(1) + 48(1)]/2

S= 49(50)/2 = 1225 (same as Gauss’ formula of n(n+1)/2)

So the sum of numbers from 50 to 150 = 11325 - 1225 = 10100 (5100 positive + 5000 negative)