The 9th term of an Arithmetic progression is 52 while the 12th term is 70. Find the product of half of first term and two-third of 5th term
We know that the nth term of an arithmetic progression can be found by the formula:
Term_n = a + (n-1)*d
where 'a' is the first term and 'd' is the common difference.
Given that:
Term_9 = a + 8*d = 52 -----> (equation 1)
Term_12 = a + 11*d = 70 -----> (equation 2)
We can solve these two equations to find 'a' and 'd'. Subtract equation 1 from equation 2 to get:
3*d = 18 --> d = 6
Substitute d = 6 into equation 1 to find 'a' :
a + 8*6 = 52 --> a = 52 - 48 = 4
We want to find the product of half of first term and two-thirds of fifth term. That is, (1/2)*a * (2/3)*Term_5.
Substituting a = 4 and Term_5 = a + 4*d = 4 + 4*6 = 28, we find:
Product = (1/2)*4 * (2/3)*28 = 2 * (56/3) = 112/3 = 37.33
This is the required product.
To find the product of half of the first term and two-thirds of the fifth term, we first need to find the values of the first term and the fifth term in the arithmetic progression.
Let's assume that the first term of the arithmetic progression is denoted by 'a', and the common difference between consecutive terms is denoted by 'd'.
We are given that the 9th term is 52. Using the formula for the nth term of an arithmetic progression, we can write:
9th term = a + (9 - 1) * d = a + 8d = 52
Similarly, we are given that the 12th term is 70. Using the same formula, we can write:
12th term = a + (12 - 1) * d = a + 11d = 70
Now we have a system of two equations with two unknowns (a and d). We can solve these equations simultaneously to find the values of a and d.
Equation 1: a + 8d = 52
Equation 2: a + 11d = 70
Subtracting Equation 1 from Equation 2, we get:
( a + 11d ) - ( a + 8d ) = 70 - 52
Simplifying, we have:
3d = 18
Dividing both sides by 3, we find:
d = 6
Substituting the value of d back into Equation 1, we can solve for a:
a + 8(6) = 52
a + 48 = 52
a = 52 - 48
a = 4
So, the first term of the arithmetic progression is 4, and the common difference is 6.
Now we can find the value of the fifth term by substituting n = 5 into the formula for the nth term:
5th term = a + (5 - 1) * d = 4 + 4 * 6 = 4 + 24 = 28
Finally, we can calculate the required product:
Product = (1/2) * first term * (2/3) * 5th term
= (1/2) * 4 * (2/3) * 28
= 2 * 2 * 4
= 16
Therefore, the product of half of the first term and two-thirds of the fifth term is 16.
To find the solution to this problem, let's first determine the common difference of the arithmetic progression.
The formula for finding the nth term of an arithmetic progression is given by:
an = a1 + (n-1)d
where "an" represents the nth term, "a1" is the first term, "n" is the position of the term, and "d" is the common difference.
We are given that the 9th term (a9) is 52 and the 12th term (a12) is 70.
Substituting these into the formula, we have:
a9 = a1 + (9-1)d (1)
a12 = a1 + (12-1)d (2)
From equation (1), we have:
52 = a1 + 8d (3)
From equation (2), we have:
70 = a1 + 11d (4)
Now, we can solve these two equations (3) and (4) simultaneously to find the values of "a1" and "d".
Subtracting equation (3) from equation (4), we have:
18 = 3d
Dividing both sides by 3, we get:
d = 6
Substituting the value of "d" into equation (3), we have:
52 = a1 + 8*6
52 = a1 + 48
a1 = 52 - 48
a1 = 4
So, the first term (a1) is 4, and the common difference (d) is 6.
Now, we can find the fifth term (a5) using the formula:
a5 = a1 + (5-1)d
a5 = 4 + 4*6
a5 = 4 + 24
a5 = 28
Finally, we can calculate the product of half of the first term and two-thirds of the fifth term:
Product = (1/2) * a1 * (2/3) * a5
Product = (1/2) * 4 * (2/3) * 28
Product = 2 * 4 * 2 * 14
Product = 8 * 28
Product = 224
Therefore, the product of half of the first term and two-thirds of the fifth term is 224.