The 4 term of an A.P is 96 while the 25 term is 173.find the product of 6 and 13 term
Let's call the first term of the arithmetic progression (AP) "a" and the common difference "d".
We are given that the 4th term is 96:
a + 3d = 96 ...(1)
We are also given that the 25th term is 173:
a + 24d = 173 ...(2)
Let's subtract equation (1) from equation (2) to eliminate "a":
(a + 24d) - (a + 3d) = 173 - 96
21d = 77
d = 77/21
d = 3.6667 (approx.)
Now substitute the value of "d" back into equation (1) to find "a":
a + 3(3.6667) = 96
a + 11 = 96
a = 85
So the first term of the AP is 85, and the common difference is 3.6667.
The 6th term of the AP is:
a + 5d = 85 + 5(3.6667) = 85 + 18.3335 = 103.3335
The 13th term of the AP is:
a + 12d = 85 + 12(3.6667) = 85 + 44 = 129
The product of the 6th and 13th terms is:
103.3335 * 129 = 13319.5007
Rounded to the nearest whole number, the product of the 6th and 13th terms is 13320.
To find the product of the 6th and 13th term, we need to determine the common difference (d) of the arithmetic progression (A.P).
Given:
The 4th term (a4) = 96
The 25th term (a25) = 173
Using the formula for the nth term of an A.P:
an = a1 + (n-1)d
We can find the value of 'a1' by substituting the values into the formula:
a4 = a1 + (4-1)d
96 = a1 + 3d
Similarly, we can find the value of 'a25':
a25 = a1 + (25-1)d
173 = a1 + 24d
Now we have a system of equations:
96 = a1 + 3d
173 = a1 + 24d
Subtracting the first equation from the second equation, we can eliminate 'a1' and solve for 'd':
173 - 96 = a1 + 24d - (a1 + 3d)
77 = 21d
d = 77/21
d = 11/3
Now that we have the value of 'd', we can find the values of the 6th (a6) and 13th term (a13), using the formula again:
a6 = a1 + (6-1)d
a6 = a1 + 5d
a13 = a1 + (13-1)d
a13 = a1 + 12d
To find the product of the 6th and 13th term, we simply multiply these two terms:
Product = a6 * a13
Now, substitute the value of 'd' into the equations to find the values of 'a6' and 'a13':
a6 = a1 + 5(11/3)
a6 = a1 + 55/3
a13 = a1 + 12(11/3)
a13 = a1 + 44
Finally, we can find the product:
Product = (a1 + 55/3) * (a1 + 44)
To find the product of the 6th and 13th term of an arithmetic progression (A.P.), we need to find the common difference (d) and then calculate the individual terms.
Step 1: Find the common difference (d):
Using the formula for the nth term of an A.P.:
a_n = a + (n-1)d
We know the 4th term (a_4) is 96 and the 25th term (a_25) is 173.
a_4 = a + (4-1)d (Equation 1)
a_25 = a + (25-1)d (Equation 2)
Subtracting Equation 1 from Equation 2 to eliminate 'a', we get:
a_25 - a_4 = a + (25-1)d - a - (4-1)d
173 - 96 = 24d
77 = 24d
Now, divide both sides by 24:
77/24 = 24d/24
3.208333 = d
So, the common difference (d) is approximately 3.208333.
Step 2: Calculate the 6th term (a_6):
Using the formula a_n = a + (n-1)d, we can substitute the values:
a_6 = a + (6-1)d
= a + 5d
Step 3: Calculate the 13th term (a_13):
Using the same formula as above, we can substitute the values:
a_13 = a + (13-1)d
= a + 12d
Step 4: Calculate the product of the 6th and 13th term (a_6 * a_13):
(a + 5d) * (a + 12d)
Let's simplify this expression:
(a + 5d) * (a + 12d)
= (a * a) + (5d * a) + (a * 12d) + (5d*12d)
= a^2 + 5ad + 12ad + 60d^2
= a^2 + 17ad + 60d^2
So, the product of the 6th and 13th term is a^2 + 17ad + 60d^2.
Please note that without the actual values of 'a' and 'd', we cannot provide the specific numeric result of the product.