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# Sequences

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In the sequence of numbers 3, 5, 2, ... each term after the first two is calculated as the term preceding it minus the term preceding that. So, for example, the third term is the second term minus the first term, i.e. 2 = 5 - 3. What is the sum of the first \(1616\) terms of the sequence?

Apr 16, 2022

#1
+23227
+1

If you write out several of the terms:   3, 5, 2, -3, -5, -2,    3, 5, 2, -3, -5, -2,    3, 5, 2, ...

you can notice that the sum of the first six terms is 0,

the sum of the next six terms is 0, etc.

Dividing 1616 by 6 gives a quotient of 269 and a remainder of 2.

This means that there are 269 groups each of which sum to 0

and two remaining terms 3 and 5 which sum to 8.

The sum of all thee terms is 8.

Apr 16, 2022

#1
+23227
+1

If you write out several of the terms:   3, 5, 2, -3, -5, -2,    3, 5, 2, -3, -5, -2,    3, 5, 2, ...

you can notice that the sum of the first six terms is 0,

the sum of the next six terms is 0, etc.

Dividing 1616 by 6 gives a quotient of 269 and a remainder of 2.

This means that there are 269 groups each of which sum to 0

and two remaining terms 3 and 5 which sum to 8.

The sum of all thee terms is 8.

geno3141 Apr 16, 2022