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.
10/2 (2a+9d) = 220
Now, the odd terms are
a, a+2d, a+4d, ...
So they form a 5-term AP whose common difference is 2d, and whose sum is
5/2 (2a+4(2d)) = 100
Now you can find a and d.
To find the first term and common difference of an arithmetic progression, we can use the formulas:
𝑆𝑛 = 𝑛/2 (2𝑎 + (𝑛 − 1)𝑑)
where 𝑆𝑛 represents the sum of 𝑛 terms, 𝑎 represents the first term, and 𝑑 represents the common difference.
In this case, we have:
𝑆10 = 220 and 𝑆odd = 100
To find the first term and common difference, we need a system of equations. We will set up two equations using the given information and solve them simultaneously.
Equation 1: 𝑆10 = 220
Substituting the values into the formula:
220 = 10/2 (2𝑎 + (10 − 1)𝑑)
Simplifying the equation:
220 = 5 (2𝑎 + 9𝑑)
Dividing both sides by 5:
44 = 2𝑎 + 9𝑑
Equation 2: 𝑆odd = 100
Using the formula for the sum of odd terms, we have:
100 = 5/2 (2𝑎 + (5 − 1)𝑑)
Simplifying the equation:
100 = 5/2 (2𝑎 + 4𝑑)
Dividing both sides by 5/2:
40 = 2𝑎 + 4𝑑
Now we have a system of equations:
Equation 1: 44 = 2𝑎 + 9𝑑
Equation 2: 40 = 2𝑎 + 4𝑑
To solve the system, we can use the method of substitution or elimination.
Let's use the method of elimination to solve the system.
Multiplying Equation 2 by (-9/2):
-9/2 * 40 = -9/2 (2𝑎 + 4𝑑)
-180/2 = -18𝑎 - 36𝑑
-90 = -18𝑎 - 36𝑑 (Equation 3)
Now, subtract Equation 3 from Equation 1:
44 - (-90) = 2𝑎 + 9𝑑 - (-18𝑎 - 36𝑑)
134 = 20𝑎 + 45𝑑
Simplifying the equation:
20𝑎 + 45𝑑 = 134
Now, we have a system of equations:
20𝑎 + 45𝑑 = 134 (Equation 4)
2𝑎 + 9𝑑 = 44 (Equation 5)
We can now solve this system of equations to find the values of 𝑎 (first term) and 𝑑 (common difference).
One way to solve this system is by multiplying Equation 5 by 5:
5 * (2𝑎 + 9𝑑) = 5 * 44
10𝑎 + 45𝑑 = 220 (Equation 6)
Now, we can subtract Equation 6 from Equation 4:
(20𝑎 + 45𝑑) - (10𝑎 + 45𝑑) = 134 - 220
10𝑎 = -86
Dividing both sides by 10:
𝑎 = -8.6
Substituting the value of 𝑎 into Equation 5:
2(-8.6) + 9𝑑 = 44
-17.2 + 9𝑑 = 44
Adding 17.2 to both sides:
9𝑑 = 61.2
Dividing both sides by 9:
𝑑 = 6.8
Therefore, the first term (𝑎) is -8.6 and the common difference (𝑑) is 6.8.