The sum of the first 10 term of an arithmetic sequence is 145 and the sum of the fourth and ninth term is five times the third term. Determine the first term and constant difference.
10/2 (2a+9d) = 145
a+3d + a+8d = 5(a+2d)
a=1
d=3
To determine the first term and constant difference of an arithmetic sequence, we can use the given information and solve the equations step-by-step.
Let's denote the first term as "a" and the common difference as "d".
Step 1: Find the sum of the first 10 terms:
The sum of an arithmetic sequence can be found using the formula: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the first n terms.
Given: Sn = 145, n = 10.
Using the formula: 145 = (10/2)(2a + (10-1)d)
Simplifying: 145 = 5(2a + 9d)
Step 2: Find the relationship between the fourth term, ninth term, and third term:
Given: The sum of the fourth and ninth term is five times the third term.
This can be written as: a + 3d + a + 8d = 5(a + 2d)
Simplifying: 2a + 11d = 5a + 10d
Step 3: Solve the system of equations obtained from step 1 and step 2:
From step 1: 145 = 5(2a + 9d) ---> 29 = 2a + 9d (Equation 1)
From step 2: 2a + 11d = 5a + 10d ---> d = 3a (Equation 2)
Substituting Equation 2 into Equation: 2a + 11(3a) = 5a + 10(3a)
Simplifying: 2a + 33a = 5a + 30a
35a = 35a
This equation is satisfied for all values of "a". Therefore, we cannot determine a unique value for the first term.
However, the relationship between the first term and common difference is given by d = 3a.
So, we have multiple solutions where the first term and common difference are related by this relationship, but the exact values cannot be determined without additional information.
To determine the first term and constant difference of the arithmetic sequence, let's approach it step by step.
Step 1: Setting up the equations.
Let's denote the first term of the arithmetic sequence as 'a' and the common difference as 'd.'
The given information can be represented by two equations:
1) The sum of the first 10 terms is 145:
S = 10/2 * (2a + (10-1)d) = 145
2) The sum of the fourth and ninth terms is five times the third term:
4th term + 9th term = 5 * 3rd term
(a + 3d) + (a + 8d) = 5(a + 2d)
Step 2: Simplifying the equations.
Let's simplify the two equations from Step 1:
1) The sum of the first 10 terms equation:
Simplifying the arithmetic series formula (using the formula for the sum of an arithmetic sequence):
10/2 * (2a + (10-1)d) = 145
5 * (2a + 9d) = 145
10a + 45d = 145
2) The sum equation for the fourth and ninth terms:
(a + 3d) + (a + 8d) = 5(a + 2d)
2a + 11d = 5a + 10d
11d - 10d = 5a - 2a
d = 3a
Step 3: Solving the equations.
Now we have two equations:
10a + 45d = 145
d = 3a
Substitute the value of 'd' from the second equation into the first equation:
10a + 45(3a) = 145
10a + 135a = 145
145a = 145
a = 1
Substitute the value of 'a' back into the second equation to find the value of 'd':
d = 3 * 1
d = 3
Therefore, the first term of the arithmetic sequence (a) is 1, and the constant difference (d) is 3.