The sum of the 1st nine terms of an arithmetic series is 216. The 1st,3rd and the 7th terms of series form the 1st three terms of a geometric series. Find the 1st term and the constant difference of the arithmetic series ?
just write what they told you:
9/2 (2a+8d) = 216
(a+2d)/(a) = (a+6d)/(a+2d)
Now solve for a and d.
Now, just for precision, you mean arithmetic sequence. A series is the sequence of partial sums of a sequence. For example,
sequence: 3,7,11,15,19,...
series: 3,10,21,36,55,...
The third term of an arithmetic sequence is 14 and the ninth term is -1 find the first four term of the sequence
To find the 1st term and the constant difference of the arithmetic series, we can use the following steps:
Step 1: Find the common difference (d) of the arithmetic series.
- Given that the sum of the first nine terms is 216, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(2a + (n-1)d), where Sn is the sum of n terms, a is the first term, and d is the common difference.
- Plugging in the known values:
216 = (9/2)(2a + (9-1)d)
216 = 4.5(2a + 8d)
216 = 9a + 36d [dividing both sides by 4.5]
24 = 9a + 36d
Step 2: Use the fact that the 1st, 3rd, and 7th terms form a geometric series.
- A geometric series has a common ratio (r) between consecutive terms.
- We can find the common ratio using the given terms:
a, a + 2d, a + 6d
- The ratio between the 2nd and 1st terms is (a + 2d) / a = r.
- The ratio between the 3rd and 2nd terms is (a + 6d) / (a + 2d) = r.
- Setting these two ratios equal to each other:
(a + 2d) / a = (a + 6d) / (a + 2d)
Cross multiplying gives:
(a + 2d)(a + 2d) = a(a + 6d)
a^2 + 4ad + 4d^2 = a^2 + 6ad
- Subtracting a^2 from both sides and simplifying:
2ad + 4d^2 = 0
2d(a + 2d) = 0
- Since d cannot be zero, the only solution is a + 2d = 0.
Thus, a = -2d.
Step 3: Substitute the value of a = -2d into the equation from Step 1 to solve for d.
24 = 9a + 36d
24 = 9(-2d) + 36d
24 = -18d + 36d
- Combine like terms:
24 = 18d
- Divide both sides by 18:
d = 24 / 18
d = 4/3
Step 4: Find the first term (a) using a = -2d.
a = -2(4/3)
a = -8/3
Therefore, the first term of the arithmetic series is -8/3 and the constant difference is 4/3.
To solve this problem, let's break it down step by step:
Step 1: Find the common difference (d) of the arithmetic series.
Let's assume that the first term of the arithmetic series is 'a' and the common difference is 'd'.
The sum of an arithmetic series can be expressed as (n/2) * [2a + (n - 1)d], where 'n' is the number of terms.
Given that the sum of the first nine terms is 216, we can write the equation as:
(9/2) * [2a + (9 - 1)d] = 216.
Simplifying the equation:
(9/2) * [2a + 8d] = 216.
Divide both sides by 9:
[2a + 8d] = 24.
Divide both sides by 2:
a + 4d = 12. ----(Equation 1)
Step 2: Formulate equations using the first, third, and seventh terms:
Given that the first, third, and seventh terms of the arithmetic series also form the first three terms of a geometric series, we can create equations using these terms.
The first term of the geometric series is 'a', and the common ratio is 'r'.
Equation 2:
a * r^(1-1) = a.
Equation 3:
a * r^(3-1) = a + 2d.
Equation 4:
a * r^(7-1) = a + 6d.
Step 3: Solve the equations simultaneously.
Now, we will solve equations 1, 2, 3, and 4 simultaneously to find the values of 'a', 'd', and 'r'.
Equating equations 2 and 3, we have:
a = a + 2d.
This implies:
2d = 0.
Thus, d = 0. ----(Equation 5)
Substituting the value of 'd' in equation 1, we have:
a + 4 * 0 = 12.
This implies:
a = 12. ----(Equation 6)
Substituting the values of 'a' and 'd' in equation 4, we have:
12 * r^6 = 12.
This implies:
r^6 = 1.
Taking the 6th root of both sides, we have:
r = ±1.
Step 4: Find the first term and the constant difference.
Since the common ratio cannot be -1 (as it is not mentioned in the question and would result in the same terms), we consider r = 1.
Therefore, the first term of the arithmetic series, 'a', is 12 (from Equation 6), and the common difference, 'd', is 0 (from Equation 5).
Hence, the first term and the constant difference of the arithmetic series are 12 and 0, respectively.