The sim of the first 10th terms of an AP is 130. If the fifth term is 3 times the first term. Find
Common difference
First term
Number of term of AP if the last term is 28
Let's denote the common difference of the arithmetic progression as 'd' and the first term as 'a'.
We are given that the sum of the first 10 terms of the AP is 130:
S10 = 130
We are also given that the fifth term is 3 times the first term:
a + 4d = 3a
Now, let's solve the equation a + 4d = 3a to find the common difference 'd':
4d = 2a
d = 0.5a
Substituting the value of 'd' in the sum formula, we have:
S10 = (10/2)(2a + 9d)
130 = 5(2a + 9(0.5a))
130 = 5(2a + 4.5a)
130 = 5(6.5a)
130 = 32.5a
Solving for 'a', we get:
a = 130/32.5
a = 4
Therefore, the first term is 4.
Since d = 0.5a, the common difference is 0.5 * 4 = 2.
Now, to find the number of terms if the last term is 28, we can use the formula for the nth term of an arithmetic progression:
an = a + (n-1)d
Substituting the known values, we have:
28 = 4 + (n-1)(2)
28 - 4 = 2n - 2
24 = 2n - 2
26 = 2n
n = 13
Therefore, the number of terms in the AP is 13.
To find the common difference, first term, and number of terms of an arithmetic progression (AP) given the sum of the first 10 terms and the relationship between the first and fifth terms, follow these steps:
Step 1: Use the sum of the first 10 terms formula:
S = (n/2)(2a + (n-1)d), where S is the sum, n is the number of terms, a is the first term, and d is the common difference.
Given that the sum of the first 10 terms is 130:
130 = (10/2)(2a + (10-1)d)
Step 2: Use the relationship between the first and fifth terms:
If the fifth term is 3 times the first term, you can express this as:
a + 4d = 3a
Step 3: Solve the system of equations formed by Step 1 and Step 2.
From Step 2, rewrite the equation as:
4d = 2a
Replace 2a with 4d in the equation from Step 1:
130 = (10/2)(4d + (10-1)d)
Simplify the equation:
130 = 5(4d + 9d)
130 = 65d
Divide both sides by 65:
d = 2
Step 4: Substitute the value of d back into the equation from Step 2 to find the first term, a:
4(2) = 2a
8 = 2a
Divide both sides by 2:
a = 4
Step 5: To find the number of terms, use the formula:
a + (n-1)d = last term
Given that the last term is 28:
4 + (n-1)(2) = 28
Simplify the equation:
(n-1)(2) = 24
Divide both sides by 2:
n-1 = 12
Add 1 to both sides:
n = 13
Therefore, the common difference is 2, the first term is 4, and the number of terms if the last term is 28 is 13 in the arithmetic progression (AP).
Let's solve the problem step by step.
Step 1: Finding the common difference (d)
To find the common difference, we need to find the difference between any two consecutive terms. In this case, we are given that the fifth term is 3 times the first term. Therefore, the difference between the fifth and first term is (3 - 1) = 2 times the common difference.
d * 4 = 2d
Step 2: Finding the first term (a)
We can express the first term (a) in terms of the common difference (d) using the given information that the fifth term is 3 times the first term.
a + 4d = 3a
Subtracting 'a' from both sides, we get:
4d = 2a
Step 3: Finding the common difference (d) and first term (a) equations
We have two equations from Step 1 and Step 2:
Equation 1: 2d = 4d
Equation 2: 4d = 2a
Step 4: Solving the equations
From Equation 1, we get:
2d = 0
This implies that the common difference (d) is equal to zero. However, this cannot be true because an arithmetic progression (AP) always has a non-zero common difference. Therefore, there is no valid solution for the given information.
So, we cannot find the common difference or the first term in this scenario.
However, if we are given that the common difference (d) is non-zero, we can proceed with the calculations as follows:
Step 5: Finding the number of terms (n) if the last term is 28
The formula to find the sum of the first 'n' terms of an arithmetic progression (AP) is:
Sn = (n/2)(2a + (n-1)d)
Where Sn represents the sum of 'n' terms, a is the first term, and d is the common difference.
We are given that the sum of the first 10 terms is 130. Plugging in the values:
130 = (10/2)(2a + (10-1)d)
130 = 5(2a + 9d)
26 = 2a + 9d
We are also given that the last term is 28. Using the formula for the 'n'th term of an AP:
an = a + (n-1)d
Plugging in the values:
28 = a + (n-1)d
Step 6: Solving the equations
We have two equations:
Equation 1: 26 = 2a + 9d
Equation 2: 28 = a + (n-1)d
These equations can be solved simultaneously to find the values of 'a' and 'd' and then determine the value of 'n'.
Since there isn't enough information given for Equation 1, we cannot find the values of 'a' or 'd' or solve for 'n'. Therefore, we do not have enough information to answer the last part of the question.