The sum of the 4th and 6th terms of an AP is 42. The sum of the 3rd and the 9th term of progression is 52, find the first term and the common difference of the sum of the 1st, 10th terms of the progression
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I have got the answer
"The sum of the 4th and 6th terms of an AP is 42"
---> a+3d + a+5d = 42
2a + 8d = 4s
a + 4d = 21
"The sum of the 3rd and the 9th term of progression is 52"
a+2d + a+8d = 52
2a + 10d = 52
a + 5d = 26
solve these two equations by subtracting them
etc
Please l need an answer to my question
To find the first term (a) and the common difference (d) of the arithmetic progression (AP), we can use the following steps:
Step 1: Set up equations using the given information.
Let's assume the first term (a) is the first term of the progression, and the common difference (d) is the difference between consecutive terms. We are given two sums:
1. The sum of the 4th and 6th terms of the AP is 42:
a + (a + 3d) = 42 ..........(Equation 1)
2. The sum of the 3rd and 9th terms of the AP is 52:
(a + 2d) + (a + 8d) = 52 ..........(Equation 2)
Step 2: Solve the equations simultaneously.
Now, we can solve these two equations to find the values of a and d.
Let's simplify Equation 1:
2a + 3d = 42 ..........(Equation 3)
And simplify Equation 2:
2a + 10d = 52 ..........(Equation 4)
Subtracting Equation 3 from Equation 4 will eliminate the variable 'a' and allow us to solve for 'd':
(2a + 10d) - (2a + 3d) = 52 - 42
7d = 10
d = 10/7
Step 3: Substitute the value of 'd' into Equation 1 or Equation 2 to solve for 'a'.
Using Equation 1:
2a + 3(10/7) = 42
2a + 30/7 = 42
14a + 30 = 294
14a = 294 - 30
14a = 264
a = 264/14
a = 18
Therefore, the first term (a) of the progression is 18, and the common difference (d) is 10/7.
Now, to find the sum of the 1st and 10th terms of this progression, we can use the formula for the sum of an AP:
Sn = (n/2)(2a + (n - 1)d)
Step 4: Calculate the sum of the 1st and 10th terms.
Using the formula, we substitute the values:
S1 = (1/2)(2 * 18 + (1 - 1) * (10/7))
S1 = 18
S10 = (10/2)(2 * 18 + (10 - 1) * (10/7))
S10 = 55
Therefore, the sum of the 1st and 10th terms of the progression is 55.