Question

Please help me with this question

Answer 1

9514 1404 393

Answer:

  -3/8 × (first term)

Step-by-step explanation:

The general term of an arithmetic sequence is ...

  an = a1 +d(n -1)

Then the ratio of interest is ...

  (7th term)/(9th term) = 5/8

  (a1 +6d)/(a1 +8d) = 5/8

Multiplying by 8(a1 +8d) we get ...

  8(a1 +6d) = 5(a1 +8d)

  8a1 +48d = 5a1 +40d . . . . . . eliminate parentheses

  8d = -3a1 . . . . . . . . . . . . . . subtract 40d+8a1

  d = (-3/8)a1 . . . . . . . . . . divide by 8 to find the common difference

The common difference is -3/8 times the first term.

__

Example

Let the first term be -16. Then the common difference is (-3/8)(-16) = 6. The first 9 terms of the sequence are ...

  -16, -10, -4, 2, 8, 14, 20, 26, 32

The ratio of the 7th and 9th terms is ...

  20/32 = 5/8 . . . . as required

__

In the general case, the ratio of terms would be ...

  (a1 +(-3/8a1)(7 -1))/(a1 +(-3/8a1)(9 -1)) = (1 -6(3/8))/(1 -8(3/8)) = (-10/8)/(-16/8) = 5/8