There are 20 terms in AP. The sum of the first 10terms is 55 and the sum of the last 10 terms is 355.find the first term and the common difference
The sum of the first 10terms is 55
---> (10/2)(2a + 9d) = 55
2a + 9d = 11
the sum of the last 10 terms is 355
---> the sum of all 20 terms is 355+55 = 410
10(2a + 19d) = 410
2a + 19d = 41
subtract them:
10d = 30
d = 3
sub into 2a + 9d = 11
2a + 27 = 11
2a = -16
a = -8
a = -8 , d = 3
Let's solve this step-by-step.
Step 1: Find the sum of the first term (a1) to the 10th term (a10).
Given that the sum of the first 10 terms is 55, we can use the formula for the sum of an arithmetic progression (AP):
Sn = n/2 * (2a1 + (n-1)d)
Substituting the given values:
55 = 10/2 * (2a1 + (10-1)d)
55 = 5 * (2a1 + 9d)
11 = 2a1 + 9d -- (Equation 1)
Step 2: Find the sum of the 11th term (a11) to the 20th term (a20).
Given that the sum of the last 10 terms is 355, we can use the same formula:
Sn = n/2 * (2a1 + (n-1)d)
Substituting the given values:
355 = 10/2 * (2a11 + (10-1)d)
355 = 5 * (2a11 + 9d)
71 = 2a11 + 9d -- (Equation 2)
Step 3: Solve the equations simultaneously.
To find the first term (a1) and common difference (d), we need to solve Equations 1 and 2 simultaneously.
Multiplying Equation 1 by 2 and subtracting Equation 2:
(22 - 2) = (4a1 + 18d) - (2a11 + 9d)
20 = 4a1 + 18d - 2a11 - 9d
20 = 4(a1 - a11) + 9d
Dividing both sides by 4:
5 = a1 - a11 + (9d/4) -- (Equation 3)
Step 4: Determine a relationship between a1 - a11 and the common difference.
Since we know that there are 20 terms in the arithmetic progression, we can write:
a1 - a11 = 19d
Substituting this into Equation 3:
5 = 19d + (9d/4)
Multiplying both sides by 4:
20 = 76d + 9d
29d = 20
d = 20/29
Step 5: Calculate the first term (a1).
Using Equation 1:
11 = 2a1 + 9(20/29)
11 - 180/29 = 2a1
(319 - 180)/29 = 2a1
139/29 = 2a1
a1 = 139/58
Therefore, the first term (a1) is 139/58 and the common difference (d) is 20/29.