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.