The sum of a AP is 26 and that of the next four terms is 74, Find the first term and the common difference
Solution
Let's denote the first term of the arithmetic progression (AP) as 'a' and the common difference as 'd'.
Step 1: Determine the sum of the first 5 terms of the AP.
The sum of the first n terms of an AP is given by the formula: Sn = (n/2)(2a + (n-1)d)
For the first 5 terms, n = 5.
So, S5 = (5/2)(2a + (5-1)d) = 26
Step 2: Determine the sum of the next 4 terms of the AP.
The sum of the next 4 terms is given by the formula: S9 = (9/2)(2a + (9-1)d) = 74
Step 3: Solve the equations obtained in Step 1 and Step 2 simultaneously to find the values of 'a' and 'd'.
From Step 1:
(5/2)(2a + 4d) = 26 [equation 1]
From Step 2:
(9/2)(2a + 8d) = 74 [equation 2]
Step 4: Solve equations 1 and 2.
Multiplying equation 1 by 9 and equation 2 by 5 to eliminate fractions:
9(5/2)(2a + 4d) = 9 * 26
5(9/2)(2a + 8d) = 5 * 74
45a + 90d = 234 [equation 3]
45a + 90d = 370 [equation 4]
Subtracting equation 3 from equation 4:
45a + 90d - (45a + 90d) = 370 - 234
0 = 136
Since 0 = 136 is not true, we have an inconsistent system of equations. This implies that there is no solution to the given problem.
To find the first term and the common difference of an arithmetic progression (AP) given the sum of the AP and the sum of the next four terms, you can use the following steps:
Step 1: Determine the formula for the sum of an AP.
The formula for the sum of an AP is: Sn = (n/2)(2a + (n-1)d), where Sn represents the sum of the first n terms, a is the first term, and d is the common difference.
Step 2: Set up the equations using the given information.
We are given two pieces of information:
- The sum of the AP is 26.
- The sum of the next four terms is 74.
Using the formula mentioned in Step 1, we can set up the following equations:
Equation 1: (n/2)(2a + (n-1)d) = 26
Equation 2: (4/2)(2a + 3d) = 74 (since there are four terms in the second sum)
Step 3: Simplify the equations.
Equation 1 becomes: n(2a + (n-1)d) = 52
Equation 2 becomes: 4(2a + 3d) = 74
Step 4: Solve the system of equations.
We have two equations and two variables (a and d). We can solve this system by substitution or elimination.
Let's use substitution method:
From Equation 1, we can express n in terms of a and d:
n = (52 - 2a) / (2a + (n-1)d)
Substitute this expression for n into Equation 2:
4(2a + 3d) = 74
8a + 12d = 74
Expand and simplify:
8a + 12d = 74
Now, substitute the value of a from Equation 2 into Equation 1:
(52 - 2a) / (2a + (52 - 2a - 1)d) = 26
Simplify:
(52 - 2a) / (52 - d) = 26
Multiply both sides by (52 - d):
52 - 2a = 26(52 - d)
Simplify and solve for a:
52 - 2a = 26(52 - d)
52 - 2a = 1352 - 26d
-2a + 26d = 1300
-2(a - 13d) = 1300
a - 13d = -650
Now, we have a system of equations:
8a + 12d = 74
a - 13d = -650
Solve this system of equations using any method (substitution, elimination, etc.) to find the values of a and d.
The solution for this system of equations will give you the first term (a) and the common difference (d) of the arithmetic progression.