Find five numbers in A.P. whose sum is 12.5 and the ratio of first to the last is 2 : 3.
Let's assume the first number in the arithmetic progression (A.P.) is 'a', and the common difference is 'd'.
According to the given information, the ratio of the first number to the last number is 2:3.
So, the last number in the A.P. can be represented as 3d.
We want to find five numbers in the A.P., and the sum of these numbers is 12.5.
The sum of an arithmetic progression can be calculated using the formula:
Sum = (n/2) * (2a + (n-1)d),
where n is the number of terms, 'a' is the first term, and 'd' is the common difference.
In this case, we have five terms, so n = 5.
Substituting the values, we get:
12.5 = (5/2) * (2a + 4d)
Simplifying the equation:
25 = 10a + 20d
Dividing both sides by 5:
5 = 2a + 4d
Since the ratio of the first number to the last number is 2:3, we can write another equation:
a / (3d) = 2/3
Simplifying the equation:
3a = 2 * 3d
3a = 6d
Dividing both sides by 3:
a = 2d
Now, we can substitute the value of 'a' in terms of 'd' into the equation '5 = 2a + 4d':
5 = 2(2d) + 4d
5 = 4d + 4d
5 = 8d
Dividing both sides by 8:
d = 5/8
Now we can find the value of 'a':
a = 2d = 2 * (5/8) = 10/8 = 5/4
So, the first number in the A.P. is 5/4, and the common difference is 5/8.
Now, to find the five numbers in the A.P., we can substitute the values of 'a' and 'd' into the A.P. formula:
The first number = a = 5/4
The second number = a + d = 5/4 + 5/8 = 10/8 + 5/8 = 15/8
The third number = a + 2d = 5/4 + 2(5/8) = 10/8 + 10/8 = 20/8 = 5/2
The fourth number = a + 3d = 5/4 + 3(5/8) = 10/8 + 15/8 = 25/8
The fifth number = a + 4d = 5/4 + 4(5/8) = 10/8 + 20/8 = 30/8 = 15/4
So, the five numbers in the A.P. are:
5/4, 15/8, 5/2, 25/8, and 15/4.
To find five numbers in an arithmetic progression (A.P.) with the given conditions, we need to apply the formulas for the sum of an A.P. and the nth term of an A.P.
Let's define the first term of the A.P. as 'a' and the common difference as 'd'.
1. Finding the sum:
The sum of an A.P. is given by the formula: Sn = n/2 * [2a + (n-1)d], where 'n' is the number of terms in the A.P.
Given, the sum is 12.5, so we have:
12.5 = 5/2 * [2a + 4d], since there are five terms in the A.P. (n = 5).
2. Finding the ratio of the first to the last term:
The ratio of the first term (a) to the last term (a + 4d) is given as 2:3.
So, we have the equation: a/(a + 4d) = 2/3.
Now we can solve these two equations simultaneously to find the values of 'a' and 'd'.
Multiplying the ratio equation by 3, we get:
3a = 2(a + 4d)
3a = 2a + 8d
a = 8d ----(Equation 1)
Substituting the value of 'a' from Equation 1 into the sum equation, we get:
12.5 = 5/2 * [2(8d) + 4d]
12.5 = 5/2 * (16d + 4d)
12.5 = 5/2 * 20d
12.5 = 50d
d = 12.5 / 50
d = 0.25
Now substituting the value of 'd' into Equation 1, we can find the value of 'a':
a = 8 * 0.25
a = 2
So, the first term 'a' is 2, and the common difference 'd' is 0.25.
Now, we can find the five numbers in the A.P.:
The first five terms in the A.P. are:
2, 2.25, 2.5, 2.75, 3
Therefore, the five numbers in A.P. whose sum is 12.5 and the ratio of the first to the last is 2:3 are 2, 2.25, 2.5, 2.75, and 3.
5/2 (2a+4d) = 25/2
a/(a+4d) = 2/3
a=2 d=1/4
2, 2¼, 2½, 2¾, 3, ...