# Mental Math Find the sum of the first ten terms of the sequence 3,4, 7, 11, 18, 29 ,...the answer is 517. How can I do this mentally? Thanks

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When faced with sequences - and especially if you are required to calculate mentally, carefully analyse the numbers when they have no apparent pattern.

3 + 4 = 7 (that is, the first and second term)

4 + 7 =11 (that is, the second and third term)

7 + 11 = 18 (that is, the third and fourth term)

11 + 18 = 29 and so on.

So now that there is a distinct pattern, it can progress

18 +29 = 47

29 + 47 = 76

47 + 76 = 123

76 + 123 = 199 (which is the tenth number in the sequence)

Add the first 6 numbers which were given at the start to the last four numbers = 445 + 72 = 517.

Remember that for each number in the sequence you have added the previous two, so you have done much of the adding as you went along. You can add in a way easiest for your mental addition style

(eg 7 + 7 + 29 + 29 = 72 (1st and 2nd = 3rd)

Similarly 4th and 5th = 6th, etc.

So add 7 + 7 + 29 + 29 + 123 + 123 + 199

In other words you are adding the 3rd number twice, the sixth number twice, the ninth number twice and then the tenth:

**therefore S (10) = 517 **

Since each of the terms in this sequence depend on the sum of the two previous terms, we expect that there will be a relationship to the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, ... where each number is the sum of the previous two numbers.

In fact, there is a strong relationship between them:

Term 1 = 3 = 3x1

Term 2 = 4 = 4x1

Term 3 = 7 = 3x1+4x1

Term 4 = 11 = 3x1+4x2

Term 5 = 18 = 3x2+4x3

Term 6 = 29 = 3x3+4x5

Term 7 = 47 = 3x5+4x8

Term 8 = 76 = 3x8+4x13

Term 9 = 123 = 3x13+4x21

Term 10 = 199 = 3x21+4x34

Noting that each of the multipliers to the 3 and 4 are in fact terms of the Fibonacci sequence.

This means that the sum of the first 10 terms is just the sum of the multipliers:

For the 3's, we have `1+1+1+2+3+5+8+13+21=55` , and for the 4's we have `1+1+2+3+5+8+13+21+34=88` .

The final sum then is simply

`3 times 55=3 times 50+3 times 5=150+15=165`

plus

`4 times 88=4 times 90-4 times 2 = 360-8=352`

which is

`165+352` break up into parts that are easier to add

`=(160+5)+(350+2)` now rearrange the sum

`=(160+350)+(5+2)`

`=510+7`

`=517`

**The sum is 517.**