5) The fifth term of an A.P exceeds twice the second term by 1. The tenth term exceeds twice the fourth term by 3 Find (i) the first term. (ii)common difference (iii)the sum of first twenty-five terms.
a+4d = 2(a+d)+1
a+9d = 2(a+3d)+3
solve for a and d, and then you want
S25 = 25/2 (2a+24d)
Let's solve the problem step by step.
Step 1: Given information
From the problem statement, we have the following information:
- The fifth term of an arithmetic progression (A.P) exceeds twice the second term by 1.
- The tenth term of the A.P exceeds twice the fourth term by 3.
Step 2: Find the second term
Let the second term of the A.P be 'a', and the common difference be 'd'.
According to the given information, we can form the equations:
- Fifth term = 2 * second term + 1
- Tenth term = 2 * fourth term + 3
Using the formula for the nth term of an A.P, we can express the terms in terms of 'a' and 'd':
Fifth term: a + 4d
Tenth term: a + 9d
Substituting these expressions into the equations, we get:
a + 4d = 2a + 1 ...(Equation 1)
a + 9d = 2(a + 3d) + 3 ...(Equation 2)
Step 3: Solve the equations
Simplifying Equation 1, we have:
4d - a = 1 ...(Equation 3)
Simplifying Equation 2, we have:
a + 9d = 2a + 6d + 3
3d - a = 3 ...(Equation 4)
Now we have two equations (Equation 3 and Equation 4) in two unknowns (a and d).
Adding Equation 3 and Equation 4, we get:
4d - a + 3d - a = 1 + 3
7d - 2a = 4 ...(Equation 5)
Step 4: Find the first term (a)
From Equation 5, we can isolate 'a':
7d - 2a = 4
-2a = 4 - 7d
2a = 7d - 4
a = (7d - 4)/2
Step 5: Find the common difference (d)
The common difference (d) can be found by substituting the value of 'a' into Equation 3:
4d - a = 1
4d - [(7d - 4)/2] = 1
Multiplying by 2 to eliminate the fraction, we get:
8d - 7d + 4 = 2
d + 4 = 2
d = 2 - 4
d = -2
Step 6: Find the first term (a)
Substituting the value of 'd' into Equation 3, we get:
4(-2) - a = 1
-8 - a = 1
a = -8 - 1
a = -9
Step 7: Summary of results
(i) The first term (a) is -9.
(ii) The common difference (d) is -2.
Step 8: Find the sum of the first twenty-five terms
To find the sum of the first twenty-five terms of an A.P, we can use the formula:
Sum of n terms = [n/2] * [2a + (n-1)d]
In this case, n = 25, a = -9, and d = -2.
Substituting the values into the formula, we get:
Sum of 25 terms = [25/2] * [2(-9) + (25-1)(-2)]
Sum of 25 terms = [25/2] * (-18 + 24)
Sum of 25 terms = [25/2] * 6
Sum of 25 terms = 25 * 3
Sum of 25 terms = 75
Step 9: Answer
(i) The first term is -9.
(ii) The common difference is -2.
(iii) The sum of the first twenty-five terms is 75.
To find the first term, common difference, and the sum of the first twenty-five terms, we need to first write the equations based on the given information.
Let's assume that the first term of the arithmetic progression (A.P) is 'a' and the common difference is 'd'.
(i) The fifth term of the A.P exceeds twice the second term by 1:
The second term would be 'a + d' since the common difference is 'd'.
The fifth term would be 'a + 4d' since we add '4' times the common difference 'd' to the first term 'a'.
The equation would be: 'a + 4d = 2(a + d) + 1'.
(ii) The tenth term exceeds twice the fourth term by 3:
The fourth term would be 'a + 3d' since we add '3' times the common difference 'd' to the first term 'a'.
The tenth term would be 'a + 9d' since we add '9' times the common difference 'd' to the first term 'a'.
The equation would be: 'a + 9d = 2(a + 3d) + 3'.
Now, we have two equations:
(a + 4d) = 2(a + d) + 1 -- Equation 1
(a + 9d) = 2(a + 3d) + 3 -- Equation 2
Let's solve these equations to find the values of 'a' and 'd'.
First, expand the equations:
a + 4d = 2a + 2d + 1
a + 9d = 2a + 6d + 3
Simplify them:
3d = a + 1 -- Equation 3
3d = a + 3 -- Equation 4
Now, we can solve Equations 3 and 4:
a + 1 = a + 3
1 = 3
The equations are not consistent, which means there must be an error in the given information or question. Please check the question again.