Question

find a1 for the arithmetic series if S7=287 and d=12

Answer 1

What is the distinction between \(a\) and \(S\) in your notation? I assume \(d\) is the difference between consecutive terms?

Answer 2

S= the sum of the sequence. D= the ratio. A1 is the first term in the sequence

Answer 3

Ah I see. Was S7 a typo, as opposed to S? Also, I believe you need to know how many terms there are to solve this.

Answer 4

And \(d\) is a difference not a ratio, if it is an arithmetic sequence. It would be a ratio of it were a geometric sequence.

Answer 5

there are 7 terms. the 7th term, or S7 = 287. youre right, d= difference, i thought they were the same.

Answer 6

Let us generalize our sequence as follows, which as seven terms. Below I write the same sums in reverse order.\[S=a_1+(a_1+d)+(a_1+2d)+\cdots+(a_1+6d)\]\[S=(a_1+6d)+(a_1+5d)+(a_1+4d)+\cdots+a_1\]Now, note the cancellation that occurs when we add the two equations together.\[2S=(2a_1+6d)+(2a_1+6d)+(2a_1+6d)+\cdots+(2a_1+6d)\]See if you can take it from there.

Answer 7

(Sorry, cancellation wasn't exactly the right term...rather the grouping that occurs when you add the two equations together. The first term on the right hand side of the third equation is the sum of the first terms on the right hand sides of the first and second equations, and so on.)