the first term of an ap is -8 the ratio of the 7th to the 9th term is 5:8 calculate the common difference of the progression
(-8+6d)/(-8+8d) = 5/8
now just solve for d
a7=a+6d...(1)a9=a+8d....(2)a7:a9=5:8a+6d/8+8d=5/8(a=8) 64+48d=40-48d 24=-8d d=-3
To find the common difference (d) of an arithmetic progression (AP), we can use the formula for the nth term of an AP:
aₙ = a₁ + (n - 1)d
Where:
aₙ is the nth term
a₁ is the first term
d is the common difference
n is the position of the term
In this case, we are given that:
a₁ = -8 (first term)
a₇ : a₉ = 5 : 8 (ratio of 7th to 9th term)
Let's find the values of a₇ and a₉ using the formula:
a₇ = a₁ + (7 - 1)d
a₇ = -8 + 6d
a₉ = a₁ + (9 - 1)d
a₉ = -8 + 8d
Now, we can use the ratio of a₇ to a₉ to create an equation:
a₇ / a₉ = 5 / 8
Substituting the values of a₇ and a₉:
(-8 + 6d) / (-8 + 8d) = 5 / 8
Cross-multiplying, we get:
8(-8 + 6d) = 5(-8 + 8d)
-64 + 48d = -40 + 40d
Now, let's solve for d:
48d - 40d = -40 + 64
8d = 24
d = 24 / 8
d = 3
Therefore, the common difference (d) of the arithmetic progression is 3.
To find the common difference of an arithmetic progression (AP), you can use the formula:
nth term = a + (n - 1) * d
Where:
- nth term represents the term you want to find
- a is the first term of the AP
- n is the position of the term in the sequence
- d is the common difference
Given that the first term (a) is -8, we can substitute this value into the formula:
-8 + (n - 1) * d
Since we know the ratio of the 7th term to the 9th term is 5:8, we can set up the following equation:
7th term / 9th term = 5/8
Substituting the formula for the nth term, we get:
(-8 + (7 - 1) * d) / (-8 + (9 - 1) * d) = 5/8
Simplifying the equation further:
(-8 + 6d) / (-8 + 8d) = 5/8
Cross-multiplying:
(8 * (-8 + 6d)) = (5 * (-8 + 8d))
Solving for d:
-64 + 48d = -40 + 40d
Rearranging terms:
48d - 40d = 64 - 40
8d = 24
Dividing both sides by 8:
d = 24/8
d = 3
Therefore, the common difference of the arithmetic progression is 3.