An arithmetic progression has 10 terms. Sum of the 10 terms is 220. Sum of the odd terms is 100. Find the first term and common difference. pl give me the answer.
10/2 (2a+9d) = 220
5/2 (2a+4*2d) = 100
To find the first term and common difference of an arithmetic progression (AP), we can use the formulas for the sum of the terms.
Let's assume the first term of the AP is 'a' and the common difference is 'd'.
Since the sum of 10 terms is given as 220, we can use the formula:
Sum = (n/2) * (2a + (n-1)d)
Substituting the given values, we get:
220 = (10/2) * (2a + 9d)
220 = 5(2a + 9d)
44 = 2a + 9d ...........(Equation 1)
Now, we know that the sum of the odd terms is given as 100. In an AP, the odd terms are 1st, 3rd, 5th, etc.
Using the formula for the sum of odd terms in an AP, we have:
Sum of odd terms = (n/2) * (2a + (n-1)d) ..... (1)
Substituting the given values, we get:
100 = (5/2) * (2a + (5-1)d)
100 = (5/2) * (2a + 4d)
100 = 5a + 10d ...........(Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) with two variables (a and d). We can solve this system to find the values of 'a' and 'd'.
Let's solve the system of equations:
Equation 1: 44 = 2a + 9d
Equation 2: 100 = 5a + 10d
By solving these equations simultaneously, we can find the values of 'a' and 'd'.
Subtracting Equation 2 from Equation 1, we get:
44 - 100 = 2a + 9d - (5a + 10d)
-56 = -3a - d
Now, we can express 'd' in terms of 'a':
d = -56 + 3a
Substituting this value of 'd' in Equation 2, we get:
100 = 5a + 10(-56 + 3a)
100 = 5a - 560 + 30a
600 = 35a
a = 600/35
a = 17.14 (approx)
Now, substituting the value of 'a' in Equation 1, we can find 'd':
44 = 2(17.14) + 9d
44 = 34.28 + 9d
9d = 44 - 34.28
9d = 9.72
d = 9.72/9
d = 1.08 (approx)
Therefore, the first term (a) is approximately 17.14 and the common difference (d) is approximately 1.08.
Please note that these values are rounded to two decimal places.
sir,
I get the answer
First term = 18.4 common difference = 4/5
is it correct ?