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
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.
To find the common difference (d) of the arithmetic progression (AP), we can use the given information about the 7th and 9th terms.
Step 1: Identify the given information.
- First term (a1) = -8
- Ratio of the 7th term to the 9th term = 5:8
Step 2: Write down the formulas.
- General term formula: an = a1 + (n-1)d, where n is the term number.
- Ratio formula: a7/a9 = 5/8
Step 3: Substitute the given values into the formulas.
Using the general term formula for the 7th and 9th terms:
a7 = -8 + 6d
a9 = -8 + 8d
Using the ratio formula:
(a7/a9) = (5/8)
Step 4: Solve the equations.
Substitute the expressions for a7 and a9 into the ratio formula:
(-8 + 6d) / (-8 + 8d) = 5/8
Cross-multiply to get rid of the fractions:
-64 + 48d = -40 + 40d
Combine like terms:
48d - 40d = -40 + 64
8d = 24
Step 5: Solve for the common difference (d).
Divide both sides of the equation by 8:
8d/8 = 24/8
d = 3
Therefore, the common difference of the arithmetic progression is 3.
To calculate the common difference of an arithmetic progression (AP), you need to use the given information about the terms of the AP. Here's how you can find the common difference:
1. Identify the first term (a) and its given value. In this case, the first term is given as -8.
2. Determine the positions of the terms involved in the given ratio (5:8). The ratio given is the 7th term to the 9th term.
3. Assign variables to the terms involved in the ratio. Let's call the 7th term "t7" and the 9th term "t9".
4. Use the formula for the nth term of an AP to express the values of t7 and t9 in terms of the first term and the common difference. The formula is: tn = a + (n-1)d, where tn represents the nth term, a is the first term, n is the position of the term, and d is the common difference.
For t7:
t7 = a + (7-1)d
t7 = -8 + 6d
t7 = -8 + 6d
For t9:
t9 = a + (9-1)d
t9 = -8 + 8d
t9 = -8 + 8d
5. Set up the equation based on the given ratio. Since the ratio of t7 to t9 is 5:8, we can write it as:
t7 / t9 = 5/8
Substitute the calculated expressions for t7 and t9 into the equation:
(-8 + 6d) / (-8 + 8d) = 5/8
6. Solve the equation for the common difference (d). Multiply both sides of the equation by (8d - 8) to eliminate the denominators and simplify the equation:
-8(8d - 8) + 6d(8d - 8) = 5(8d - 8)
-64d + 64 + 48d^2 - 48d = 40d - 40
48d^2 - 152d + 104 = 40d - 40
48d^2 - 152d - 40d + 144 = 0
48d^2 - 192d + 144 = 0
Divide the entire equation by 8 to simplify:
6d^2 - 24d + 18 = 0
3d^2 - 12d + 9 = 0
Factorize the quadratic equation:
(3d - 3)(d - 3) = 0
From this equation, we can see that either (3d - 3) = 0 or (d - 3) = 0.
Solving for d, we have two solutions:
3d - 3 = 0 => 3d = 3 => d = 1
d - 3 = 0 => d = 3
Therefore, the possible common differences for the arithmetic progression are d = 1 and d = 3.