Question

Each level of a smartphone app adds more gems for you to match. On level one, there were 13 gems. On level twelve, there were 123 gems. When you complete all twelve levels, how many total gems will you have matched?

Answer 1

So have you found any correlation between the number of gems and the level?

Answer 2

@AllTehMaffs no i havent.. i have been searching for it though

Answer 3

^_^ hint: look at the number that precede "3" in each sequence - what is it for level 1?

Answer 4

~ in the number of jewels, I'm sorry.

Like, for level 1, if we say that g is the number of gems, g=13 . What number is in the tens digit?

Answer 5

@AllTehMaffs i dont get what you're saying..

Answer 6

@AllTehMaffs ohh 3

Answer 7

the 3 is in the "ones" digit. 1 is in the tens digit ^_^

Answer 8

So for level 1, it's

10 + 3 gems

Answer 9

@AllTehMaffs oh yea you're right. my mistake

Answer 10

how did you get the 10?

Answer 11

13 = 10 + 3

Answer 12

i get that lol. but im saying how would you know how much to add for each level?

Answer 13

and the answer is 816? correct?

Answer 14

Saying that another way, for level 1

\[ 13 = 10+3 = (10*1 + 3 )\text{ gems}\]

So then for level 12, there are 123 gems, so we can say there are

\[123 = 120 + 3 = 12*10 + 3 \text{ gems}\]

Can you see the pattern those two numbers followed?

Answer 15

yup, that's the answer - how did you get it?

Answer 16

The pattern is 10 times the level plus three gems

Level 4 would be 43 gems
Level 3 would be 33 gems
Level 8 = 83 gems, etc.

Answer 17

If we look at the sums of the sequence then, it's tedious just to add each term individually. But if we add the sequence to itself, we can see some nice symmetry!

\[\tiny \begin{array}{l}
13 + &23 + &33 + &43 + &53 + &63 + &73 + &83 + &93 + &103 + &113 + &123 =& ?
\\ 123 +&113 +&103 +&93 +&83 +&73 +&63 +&53 +&43 +&33 +&23 +&13 =& ?
\\ \hline
\\ 136 +&136 +&136 +&136 +&136 +&136 +&136 +&136 +&136 +&136 +&136 +&136 =& 12(136)
\end{array}\]

So if we add the first term and the last term, then multiply it by the number of terms in the sequence and average it, we can quickly find the sum of the sequence.

\[ \frac{n(a_1+a_n)}{2} = \text{sum of sequence}\]
\[ \frac{12(13+123)}{2} = 816\]
Just like you had.