Mental Math Find the sum of the first ten terms of the seque

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

Solution

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