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

816 gems.

Explanation:

Given that each level of a smartphone app adds more gems for you to match.

On level 1: 12 gems

On level 12: 123 gems

we have to find the total gems up to level 12th i.e sum
`S_12`

12,.........., 123

here,
`a=13, a_(12)=123`

As, nth term formula for AP is

`a_n=a+(n-1)d`

⇒
`a_12=a+(12-1)d`

⇒
`123=13+11d`

⇒ d=10

Now, we have to find the sum

`S_(12)=(12)/(2)(2a+(13-1)d)=6(2(13)+11(10))=6(26+110)=816`

Hence, 816 gems matched.

Answer 2

This is an example of an arithmetic sequence.
On level one: a 1 = 13.On level twelve: a 12 = 123.The sum of all 12 levels:S n = n/2 * ( a 1 + a n )S 12 = 12/2 * ( a 1 + a 12 )S 12 = 6 * ( 13 + 123 )S 12 = 6 * 136S 12 = 816Answer: I will have 816 gems matched.