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The first term of an ap is -8 and the ratio of the 7th term to the 9th term is 5:8
calculate the common difference of the progression
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GPT 3.5
To find the common difference of an arithmetic progression (AP), we can use the formula:
nth term = first term + (n - 1) * common difference
Let's use this formula to find the 7th and 9th terms of the AP:
7th term = -8 + (7 - 1) * common difference
9th term = -8 + (9 - 1) * common difference
The ratio of the 7th term to the 9th term is given as 5:8:
(7th term) / (9th term) = 5/8
Plugging in the formulas from above and simplifying the ratio equation:
(-8 + 6 * common difference) / (-8 + 8 * common difference) = 5/8
Cross-multiplying:
8 * (-8 + 6 * common difference) = 5 * (-8 + 8 * common difference)
-64 + 48 * common difference = -40 + 40 * common difference
Subtracting 40 * common difference from both sides and adding 64 to both sides:
64 - 40 = 40 * common difference - 48 * common difference
24 = -8 * common difference
Dividing both sides by -8:
total difference = -8/24 = -1/3
Therefore, the common difference of the arithmetic progression is -1/3.
nth term = first term + (n - 1) * common difference
Let's use this formula to find the 7th and 9th terms of the AP:
7th term = -8 + (7 - 1) * common difference
9th term = -8 + (9 - 1) * common difference
The ratio of the 7th term to the 9th term is given as 5:8:
(7th term) / (9th term) = 5/8
Plugging in the formulas from above and simplifying the ratio equation:
(-8 + 6 * common difference) / (-8 + 8 * common difference) = 5/8
Cross-multiplying:
8 * (-8 + 6 * common difference) = 5 * (-8 + 8 * common difference)
-64 + 48 * common difference = -40 + 40 * common difference
Subtracting 40 * common difference from both sides and adding 64 to both sides:
64 - 40 = 40 * common difference - 48 * common difference
24 = -8 * common difference
Dividing both sides by -8:
total difference = -8/24 = -1/3
Therefore, the common difference of the arithmetic progression is -1/3.
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