The 4th term of an A.P is 96 while the 25th term is 173 find sum of 3rd term and 6th term and product of 6th and 10th term
a + 3d = 96
a + 24d = 173
subtract them
21d = 77
d = 11/3
a + 3(11/3) = 96
a = 85
so term(6) = a + 5d = 85 + 55/3 = 310/3
in the same way, find term(3) and term(10)
then do the calculation asked for
To find the sum of the 3rd term and 6th term, we first need to find the common difference (d) of the arithmetic progression (A.P).
We know that the 4th term (a4) is 96, so we can write it as:
a4 = a1 + (4-1) * d, where a1 is the first term.
96 = a1 + 3d
Now, let's find the value of the first term (a1) in terms of d. We can use the 25th term (a25) to do this.
a25 = a1 + (25-1) * d
173 = a1 + 24d
Now, we have a system of equations:
96 = a1 + 3d
173 = a1 + 24d
By solving these two equations simultaneously, we can find the values of a1 and d.
Subtracting the first equation from the second equation, we get:
173 - 96 = (a1 + 24d) - (a1 + 3d)
77 = 21d
Dividing both sides by 21, we find:
d = 77/21 = 11/3
Substituting the value of d back into the first equation, we can find a1:
96 = a1 + 3 * (11/3)
96 = a1 + 11
Subtracting 11 from both sides, we get:
a1 = 85
Now that we know the values of a1 and d, we can find the 3rd term (a3) and the 6th term (a6):
a3 = a1 + (3-1) * d
a3 = 85 + 2 * (11/3) = 85 + 22/3 = 157/3
a6 = a1 + (6-1) * d
a6 = 85 + 5 * (11/3) = 85 + 55/3 = 260/3
Therefore, the sum of the 3rd term and 6th term is:
a3 + a6 = (157/3) + (260/3) = 417/3
To find the product of the 6th term (a6) and the 10th term (a10), we can simply multiply these two terms:
a6 * a10 = (260/3) * (a1 + (10-1) * d)
Substituting the known values of a1 and d, we can calculate the product:
a6 * a10 = (260/3) * (85 + 9 * (11/3)) = (260/3) * (85 + 99/3) = (260/3) * (85 + 33) = (260/3) * (118) = 10,360/3 = 3453.33
To find the sum of the 3rd term and the 6th term of an arithmetic progression (A.P) and the product of the 6th and 10th term, we need to find the common difference (d) and then calculate the terms using the given information.
Let's start with finding the common difference (d) first.
The formula to calculate the nth term of an arithmetic progression is:
an = a + (n - 1)d
Where:
an is the nth term
a is the first term
n is the position of the term in the sequence
d is the common difference
Given:
a4 = 96 (4th term)
a25 = 173 (25th term)
Using the formula for the 4th term, we have:
a + (4 - 1)d = 96
a + 3d = 96 ----(1)
Using the formula for the 25th term, we have:
a + (25 - 1)d = 173
a + 24d = 173 ----(2)
Now, we have two equations (1) and (2) with two unknowns (a and d).
Let's solve these equations simultaneously to find the values of a and d.
Subtracting equation (1) from equation (2), we get:
(a + 24d) - (a + 3d) = 173 - 96
21d = 77
d = 77/21
d = 3.6667 (rounded to 4 decimal places)
Substituting d back into equation (1), we can solve for a:
a + 3(3.6667) = 96
a + 11 = 96
a = 96 - 11
a = 85
Now that we have found the values of a and d, we can calculate the desired terms.
To find the 3rd term (a3), we substitute n = 3 into the formula:
a3 = a + (3 - 1)d
a3 = 85 + (3 - 1) * 3.6667
a3 = 85 + 2 * 3.6667
a3 = 85 + 7.3334
a3 = 92.3334
To find the 6th term (a6), we substitute n = 6 into the formula:
a6 = a + (6 - 1)d
a6 = 85 + (6 - 1) * 3.6667
a6 = 85 + 5 * 3.6667
a6 = 85 + 18.3335
a6 = 103.3335
To find the product of the 6th and 10th term, we simply multiply a6 and a10:
Product = a6 * a10
Product = 103.3335 * (85 + (10 - 1) * 3.6667)
Calculating a10:
a10 = a + (10 - 1)d
a10 = 85 + (10 - 1) * 3.6667
a10 = 85 + 9 * 3.6667
a10 = 85 + 33.0003
a10 = 118.0003
Product = 103.3335 * 118.0003
Product = 12196.668
Therefore, the sum of the 3rd term and the 6th term is approximately 92.3334 + 103.3335 = 195.6669, and the product of the 6th and 10th term is approximately 12196.668.