Find four consecutive terms in A.P whose sum is 72 and the ratio of product of the extreme terms to the product of means is 9:10
the four terms are
a, a+d, a+2d, a+3d
their sum = 72
4a + 6d = 72
2a + 3d = 36 **
product of extremes: a(a+3d)
product of means: (a+d)(a+2d)
a(a+3d) / (a+d)(a+2d) = 9/10
10a^2 + 30ad = 9a^2 + 27ad + 18d^2
a^2 + 3ad - 18d^2 = 0
(a + 6d)(a - 3d) = 0
a = -6d or a = 3d ***
if a = 3d, in **
2(3d) + 3d = 36
9d = 36
d = 4 , then a = 12
and the terms are 12, 16, 20, 24
if a = -6d in **
2(-6d) + 3d = 36
-9d = 36
d = -4 , a = 24
terms are : 24, 20, 16, 12
notice the terms are just in reverse order
Thanks a lot!
four consecutive terms are a-3d , a-2d, a+2d , a+3d
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To solve this problem, we need to understand the concepts of an arithmetic progression (A.P.), the sum of consecutive terms in an A.P., and the ratio of the product of the extreme terms to the product of the means.
Let's start with the first part of the problem - finding four consecutive terms in an A.P. whose sum is 72.
Step 1: Let's assume the first term of the A.P. is 'a', and the common difference is 'd'.
Step 2: The four consecutive terms in the A.P. will be 'a', 'a + d', 'a + 2d', and 'a + 3d'.
Step 3: The sum of these four consecutive terms can be calculated as:
Sum = a + (a + d) + (a + 2d) + (a + 3d) = 4a + 6d
Step 4: According to the problem statement, the sum of these four terms is 72. So, we can write the equation as:
4a + 6d = 72
Now, let's move on to the second part of the problem - the ratio of the product of the extreme terms to the product of the means.
Step 5: The product of the extreme terms (first and fourth terms) can be calculated as:
Product of extreme terms = a * (a + 3d)
Step 6: The product of the means (second and third terms) can be calculated as:
Product of means = (a + d) * (a + 2d)
Step 7: According to the problem statement, the ratio of the product of the extreme terms to the product of the means is 9:10. So, we can write the equation as:
(a * (a + 3d))/( (a + d) * (a + 2d)) = 9/10
Now, we have two equations. To find the values of 'a' and 'd', we can solve these equations simultaneously.
Step 8: Solve the equations:
Equation 1: 4a + 6d = 72
Equation 2: (a * (a + 3d))/( (a + d) * (a + 2d)) = 9/10
Solving these equations will give you the values of 'a' and 'd'. Once you have those values, you can find the four consecutive terms in the A.P. by substituting the values in the expression: 'a', 'a + d', 'a + 2d', and 'a + 3d'.
Note: The solution to the equations might result in more than one set of values for 'a' and 'd', which will give you different sets of four consecutive terms.