Three numbers in AP have sum 33 and products 1232 .Find the numbers
Let the three numbers be a-d, a, and a+d (since they are in AP)
Given that their sum is 33:
(a-d) + a + (a+d) = 33
Simplifying, we get:
3a = 33
a = 11
Now, given that their product is 1232:
(a-d) * a * (a+d) = 1232
Substituting a = 11, we get:
(11-d) * 11 * (11+d) = 1232
121 - d^2 = 112
d^2 = 9
d = 3 or -3
Therefore, the three numbers are either 8, 11, 14 or 14, 11, 8.
The sum of three numbers in AP is 30 and the sum of their squares is 398 .Find the numbers
We are given that three numbers form an arithmetic progression (AP) with a sum of 33 and a product of 1232. Let's assume that the three numbers are a-d, a, and a+d, where "a" is the middle term and "d" is the common difference.
To find the values of "a" and "d," we can use the given information:
1) Sum of the three numbers:
(a-d) + a + (a+d) = 33
3a = 33
a = 11
2) Product of the three numbers:
(a-d) * a * (a+d) = 1232
(a^2 - d^2) = 1232
Plugging in the value of "a" from equation 1:
(11^2 - d^2) = 1232
121 - d^2 = 1232
-d^2 = 1111
d^2 = -1111
Since "d^2" cannot be negative, it means that there is no common difference that satisfies our given conditions. Therefore, there is no arithmetic progression that satisfies the given sum and product.
To find the three numbers in an arithmetic progression (AP) with a given sum and product, we can use a combination of algebraic equations and substitution.
Let's assume the three numbers in AP are a-d, a, and a+d, where 'a' is the middle term, and 'd' is the common difference.
Given that the sum of the three numbers is 33, we can write the equation as:
(a - d) + a + (a + d) = 33
Simplifying this equation, we get:
3a = 33
a = 11
Now, let's move on to finding the product of the three numbers. We know that the product of the three numbers is 1232, so we can write another equation as:
(a - d) * a * (a + d) = 1232
Substituting the value of 'a' as 11, we get:
(11 - d) * 11 * (11 + d) = 1232
Now, we need to find the factors of 1232, which satisfy the equation. We can do this by trial and error or by using prime factorization.
To find the prime factorization of 1232:
1. Divide 1232 by the smallest prime number, which is 2: 1232 ÷ 2 = 616.
2. Repeat the process with the quotient (616): 616 ÷ 2 = 308.
3. Again, divide 308 by 2: 308 ÷ 2 = 154.
4. Continue this process until you can't divide any further by 2: 154 ÷ 2 = 77. Since 77 is an odd number, we can move on to the next prime number, which is 3.
5. Divide 77 by 3: 77 ÷ 3 = 25.67. Since 25.67 is not a whole number, we can move on to the next prime number, which is 5.
6. Divide 77 by 5: 77 ÷ 5 = 15.4. Again, since 15.4 is not a whole number, we can move on to the next prime number, which is 7.
7. Divide 77 by 7: 77 ÷ 7 = 11. Since 11 is a prime number, we stop here.
So, the prime factorization of 1232 is 2^4 × 7 × 11.
Now, we need to find the factors of 1232 that satisfy the equation (11 - d) * 11 * (11 + d) = 1232.
One possible combination is:
(11 - 4) * 11 * (11 + 4) = 1232
7 * 11 * 15 = 1232
Hence, the three numbers in AP are 7, 11, and 15.