Question

A sequence of 7 numbers starts with 8. The second number in the sequence is unknown and may be positive or negative. The third through seventh numbers are the sum of the previous two numbers in the sequence, and the seventh number is 0. What is the sum of the 7 numbers?

Answer 1

4

Explanation:

[TeX] a_1=8,a_7=0[/TeX]

Since the third term is the sum of the two previous terms

[TeX] a_3=a_2+a_1[/TeX]

[TeX] a_7=a_6+a_5[/TeX]

[TeX] a_7=a_5+a_4+a_4+a_3[/TeX]

Continuing in like manner

[TeX] a_7=5a_3+3a_2[/TeX]

Since [TeX] a_3=a_2+a_1[/TeX]

[TeX] a_7=5(a_2+a_1+3a_2[/TeX]

[TeX] a_7=5a_2+5a_1+3a_2[/TeX]

[TeX] a_7=8a_2+5a_1[/TeX]

Recall: [TeX] a_1=8,a_7=0[/TeX]

[TeX] 0=8a_2+5*8[/TeX]

[TeX] 8a_2=-40[/TeX]

[TeX] a_2=-5[/TeX]

The sequence is therefore:

8,-5,3,-2,1,-1,0

The sum of the seven numbers is 4.