Sum of Even Numbers & Percentages

[sarasoleja]

Q1: What is the sum of first 20 even numbers?

The answer I got was 110 ... I added (2+4+6+8+10+12+14+16+18+20) , but the answer on the answer sheet was 420. What is the trick in the question? What method can be used to find the answer? Where did I go wrong?

Q2: Mary's height increased by 20% between her 5th and 10th birthday. It increased by 15% between her 10h and 15th birthday, and it increased by 10% between her 15th and 20th birthday. By how much did her height increase between her 5th and 20th birthday?

What I did was assume her height was 100 and added 20% which gave me 120. and then took 15% of that (120) and added and then did the same with the 10% which gave me 51.8%, but my answer sheet says 50.8% ... Where did I go wrong? What tactic could be used to solve this question.

I would really appreciate ur help!

 

[Deleted member 4993]

Q1: What is the sum of first 20 even numbers?

The answer I got was 110 ... I added (2+4+6+8+10+12+14+16+18+20) , but the answer on the answer sheet was 420. What is the trick in the question? What method can be used to find the answer? Where did I go wrong?

The 20 th even number is not 20.

1st even number = 2 = 1 * 2

2nd even number = 4 = 2 * 2

3 rd even number = 6 = 2 * 3

4 th even number = 8 = 2 * 4

.
.
.
n th even number = ???


Q2: Mary's height increased by 20% between her 5th and 10th birthday. It increased by 15% between her 10h and 15th birthday, and it increased by 10% between her 15th and 20th birthday. By how much did her height increase between her 5th and 20th birthday?

What I did was assume her height was 100 and added 20% which gave me 120. and then took 15% of that (120) and added and then did the same with the 10% which gave me 51.8%, but my answer sheet says 50.8% ... Where did I go wrong? What tactic could be used to solve this question.

I am also getting 51.8% - may be the answer sheet has a typo (or we are both wrong!!)

I would really appreciate ur help!

 

[sarasoleja]

Obviously I misunderstood the first question.

and Yeah I'll ask my instructor about the 2nd Question to make sure !

Thanks a million for your kind help !

 

[masters]

Q1: What is the sum of first 20 even numbers?

The answer I got was 110 ... I added (2+4+6+8+10+12+14+16+18+20) , but the answer on the answer sheet was 420. What is the trick in the question? What method can be used to find the answer? Where did I go wrong?

I would really appreciate ur help!

Hi sarasoleja,

You only added 10 terms. The question asks "What is the sum of the first 20 terms?"

If you are familiar with arithmetic sequences and series, you could use a couple of nice formulas to help you out.

To find out what the 20th term is you could use \(\displaystyle a_n=a_1+(n-1)d\) where:

\(\displaystyle a_n\) is the nth term (in your case you would be looking for \(\displaystyle a_{20}\)

\(\displaystyle a_1\) is the first term (in your case you would use 2)

\(\displaystyle n\) is the number of terms (in your case that would be 20.

\(\displaystyle d\) is the common difference (in your case the difference between each term is 2)

So, we have \(\displaystyle a_{20}=2+(20-1)2\)

\(\displaystyle a_{20}=40\). So the last term in your sequence will be 40.

Now, on to another helpful formula....the sum of an arithmetic sequence: \(\displaystyle S_n=\frac{n(a_1+a_n)}{2}\)

For your problem: \(\displaystyle S_{20}=\frac{20(2+40)}{2}=420\)

 

[TchrWill]

Q1: What is the sum of first 20 even numbers?

The answer I got was 110 ... I added (2+4+6+8+10+12+14+16+18+20) , but the answer on the answer sheet was 420. What is the trick in the question? What method can be used to find the answer? Where did I go wrong?

Q2: Mary's height increased by 20% between her 5th and 10th birthday. It increased by 15% between her 10h and 15th birthday, and it increased by 10% between her 15th and 20th birthday. By how much did her height increase between her 5th and 20th birthday?

What I did was assume her height was 100 and added 20% which gave me 120. and then took 15% of that (120) and added and then did the same with the 10% which gave me 51.8%, but my answer sheet says 50.8% ... Where did I go wrong? What tactic could be used to solve this question.

I would really appreciate ur help!

The sum of the first "n" even numbers is S = n(n + 1).

Look at it this way:

n........1........2........3........4........5...
N........2........4........6........8.......10...
Sum....2........6........12......20.......30...
Diff.........4.........6........8.......10...
Diff..............2........2........2.

With the second differences being constant at 2, the general expression for the sum of the first n even numbersis of the form
N = an^2 + bn + c

Using the known data, we can write
a(1^2) + b(1) + c = 2 or a + b = c = 2
a(2^2) + b(2) + c = 6 or 4a + 2b + c = 6
a(3^2) + b(3) + c = 12 or 9a + 3b + c = 12

Solving, a = 1, b = 1 and c = 0 yielding N = n(n + 1)


As for the height increase:
Age..............5...........10...........15...........20
Growth %.............20...........15...........10
Height..........X..........1.2X.........1.38X......1.518X or a total growth over the 5 to 20 year old period of 51.8%

Clearly, the book is wrong.