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.
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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
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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
