The first term of an AP is-8. The ratio of the 7th term to the 9th term is 5:8. Calculate the common difference of the progression.
The first term of an arithmetic sequence is -1and the first fifteenth ters is 27 find the common difference and the sum of the first fifteen terms
To find the common difference of an arithmetic progression (AP), we can use the formula:
\[ a_n = a + (n-1)d \]
where \( a_n \) is the \( n \)th term of the AP, \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference.
Given that the first term, \( a \), is -8, we can substitute this value into the formula:
\[ a_n = -8 + (n-1)d \]
To find the common difference, we need to consider the ratio of the 7th term to the 9th term. The ratio is given as 5:8, which means that the 7th term is 5 units away from the 9th term.
Using the formula to substitute the values:
\[ a_7 = -8 + (7-1)d \]
\[ a_9 = -8 + (9-1)d \]
According to the given ratio:
\[ \frac{{a_7}}{{a_9}} = \frac{5}{8} \]
Substituting the formulas for \( a_7 \) and \( a_9 \):
\[ \frac{{-8 + (7-1)d}}{{-8 + (9-1)d}} = \frac{5}{8} \]
To simplify this equation further, let's multiply both sides by 8 to eliminate the fraction:
\[ 8(-8 + (7-1)d) = 5(-8 + (9-1)d) \]
Expanding and simplifying the equation:
\[ -64 + 56d = -40 + 40d \]
Combine like terms:
\[ 56d - 40d = -40 + 64 \]
\[ 16d = 24 \]
Dividing both sides by 16:
\[ d = \frac{24}{16} \]
\[ d = 1.5 \]
So, the common difference of the arithmetic progression is 1.5.
To calculate the common difference of an arithmetic progression (AP), we can use the formula:
nth term = a + (n - 1)d
where:
- nth term is the term at position n
- a is the first term of the AP
- d is the common difference of the AP
Given that the first term (a) of the AP is -8, we can substitute this value into the formula:
-8 + (n - 1)d
Next, we are given the ratio of the 7th term to the 9th term, which is 5:8. This means that the 7th term divided by the 9th term is equal to 5/8:
7th term / 9th term = 5/8
Substituting the formula for the nth terms into this ratio:
(-8 + (7 - 1)d) / (-8 + (9 - 1)d) = 5/8
Simplifying the equation:
(-8 + 6d) / (-8 + 8d) = 5/8
Cross-multiplying, we get:
-64 + 48d = -40 + 40d
Combining like terms:
8d = 24
Dividing both sides by 8, we find:
d = 3
Therefore, the common difference of the arithmetic progression is 3.
a = -8
(a+6d)/(a+8d) = 5/8
So, plug in a and solve for d:
(6d-8)/(8d-8) = 5/8
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