What is the total in this sequence - 5 + 10 + ....... + 550 + 555?

Question

It is a sequence that starts at 5 and adds five more each time. I need to know the sum

Answer 1

n=110. a_i=5i, i=1,..111

S₁₁₁=(a₁+a₁₁₀)(n/2)=5(1+111)111/2=5*56*111=31 080

 

Answer 2

There may be multiple ways to think about this, but I'll give my interpretation.

So first, factor out 5 from each number in the series:

 

5 (1 + 2 + 3 + ... + 109 + 110 + 111) = ?

Now let's just look at the (1 + 2 + 3 + ... + 109 +110 + 111) part.

 

Here's where you have to work a little bit of magic. Notice that

 

• 1 + 111 = 112
• 2 + 110 = 112
• 3 + 109 = 112

And so on.

 

Take a portion of 111 and give it to 1.

Take a portion of 110 and give it to 2.

Take a portion of 109 and give it to 3.

 

What portion of 111 should we give to 1? Let's try looking at the midpoint. (1 + 111) / 2 = 112 / 2 = 56.

Take 55 from 111 and give it to 1. 

 

111 - 55 = 56.

1 + 55 = 56.

 

What portion of 110 should we give to 2? Let's try looking at the midpoint. (2 + 110) / 2 = 112 / 2 = 56.

Take 54 from 110 and give it to 2.

 

110 - 54 = 56.

2 + 54 = 56.

 

Soon you will realize that you have 111 56's added up. Here's a visual that could hopefully help:

 

1 + 2 + 3 + ... + 109 + 110 + 111

56 + 2 + 3 + ... + 109 + 110 + 56 (Give 55 from 111 to 1)

56 + 56 + 3 + ... + 109 + 56 + 56 (Give 54 from 110 to 2)

56 + 56 + 56 + ... + 56 + 56 + 56 (Give 53 from 109 to 3)

 

Let's double check this method looking at both sides of 56.

 

... + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + ...

... + 52 + 53 + 54 + 56 + 56 + 56 + 58 + 59 + 60 + ... (Give 1 from 57 to 55)

... + 52 + 53 + 56 + 56 + 56 + 56 + 56 + 59 + 60 + ... (Give 2 from 58 to 54)

... + 52 + 56 + 56 + 56 + 56 + 56 + 56 + 56 + 60 + ... (Give 3 from 59 to 53)

... + 56 + 56 + 56 + 56 + 56 + 56 + 56 + 56 + 56 + ... (Give 4 from 60 to 52)

 

Checks out.

 

So we have 5 (initially factored out in the beginning) multiplied by 111 56's:

 

5(56x111) = 31,080.

Answer 3

   n

  ∑ a_i = (n/2)(a₁+a_n)

 i=1

 

Above is the formula for finding the sum of the first n terms of

an arithmetic series.

We are starting at a₁ = 5  and adding 5 repeatedly until we reach 550

 

To find n, the number of terms in the series:

Starting at 5 we add 5 a total of n-1 times to reach 550

5 + (n-1)5 = 550

5 + 5n - 5 = 550

5n = 550

 

n = 110 terms in the sequence

Check:

-5+5(109) = 5 + 545 = 550

 

  110

   ∑ a_i = (110/2)(5 + 550)

   i=1

 

Sum = 55(555) = 30,525