In the sequence S, the difference between . . .

Vincen wrote:In the sequence S, the difference between any two consecutive terms is equal. If the sum of the fourth term and the fifth term of the sequence is equal to the seventh term of the sequence, what is the value of the second term of the sequence?

A. -4
B. 0
C. 4
D. 8
E. Cannot be Determined

The OA is B.

Why is B the correct answer? How can I determine this value? I need your help experts.

Thanks.

Call the first number in the sequence N. The problem states that consecutive terms differ by a constant amount, call it X.

So the first number in the sequence is N. The second N+X and so on, with fourth term being N+3X, fifth being N+4X and seventh being N+6X.

The problem states that the sum of fourth and fifth terms equals seventh term, so:

(N+3X)+(N+4X) = N+6X

simplifying 2N+7X=N+6X yields X=-N

Looks strange but continue with that and substitute back into the second sequence term above

N+X = N+(-N) =
0, B

Hello Vincen.

We have that the difference between any two consecutive terms is equal, that is to say, $$S=a,\ \ a+d,\ \ a+2d,\ \ a+3d,\ \ a+4d,\ \ a+5d,\ \ a+6d\ .\ .\ .\ .$$ Also we have that the sum of the fourth term and the fifth term of the sequence is equal to the seventh term, that is to say, $$\left(a+3d\right)+\left(a+4d\right)=a+6d\ \leftrightarrow\ 2a+7d=a+6d\ \leftrightarrow\ \ a=-d.$$ Replacing this value of "a" in the second term of the sequence we will get $$a+d=-d+d=0.$$ So, the correct answer is B.

I hope this explanation may help you.

I'm available if you'd like a follow up.

Regards.