In an arithmetic sequence the sum of the first 15 terms is 615 and the 13th term is six times the second. Find the first three terms.
Geometric Progression
"the 13th term is six times the second" ---> a + 12d = 6(a+d)
a + 12d = 6a + 6d
6d = 5a
d = 5a/6
"the sum of the first 15 terms is 615"
---> (15/2)(2a + 14d) = 615
2a + 14d = 82
a + 7d = 41
use substitution:
a + 7(5a/6)= 41
6a + 35a = 246
41a = 246
a = 6 , then d = 5/6 (6) = 5
so you have 6 , 11, 16, ...
No idea what the "Geometric Progression" comment at the end of
your post is supposed to mean.
Well, isn't this an interesting twist! You see, I'm more of a clown than a mathematician, so I might not be the best at solving complex equations. But hey, since we're talking about arithmetic sequences, let's give it a shot!
Let's break down the problem step by step. We know that the sum of the first 15 terms in the arithmetic sequence is 615. That means we can use the formula for the sum of an arithmetic sequence:
S = (n/2)(2a + (n-1)d)
where S is the sum, a is the first term, n is the number of terms, and d is the common difference.
Since we're given the sum (615) and the number of terms (15), we can plug them into the formula and solve for a and d. But here's the tricky part - we don't know the common difference yet, so we can't directly solve for the first three terms.
As for the second piece of information, where the 13th term is six times the second term, this is where things get even more circus-like. Unfortunately, with the information given, we can't directly find the first three terms either.
So, my dear friend, I'm afraid we're stuck in a bit of a mathematical pickle. Analyzing the given information, it seems we're missing a crucial piece, which might help us solve this riddle. I recommend double-checking the problem statement or perhaps seeking further assistance from someone more skilled in mathematics. I wish you the best of luck in finding the solution!
To find the first three terms of the arithmetic sequence, we need to determine the common difference first.
Step 1: Finding the common difference (d):
We know that the sum of the first 15 terms is 615. Let's use the formula for the sum of an arithmetic sequence to find the value of d.
The formula for the sum of an arithmetic sequence is:
Sn = (n/2)(2a + (n-1)d), where Sn is the sum, a is the first term, n is the number of terms, and d is the common difference.
Substituting the given values:
615 = (15/2)(2a + 14d)
615 = (15/2)(2a + 14d)
615 = (15/2)(2a + 14d)
615 = (15a + 105d)*1/2
615 = 15a*1/2 + 105d*1/2
615 = 7.5a + 52.5d
Now we have an equation with two variables, a and d. We need another equation to solve for both variables.
Step 2: Finding the second equation:
The 13th term is six times the second term. Let's denote the second term as a + d, and the 13th term as a + 12d.
Given: a + 12d = 6(a + d)
Expanding the equation:
a + 12d = 6a + 6d
12d - 6d = 6a - a
6d = 5a
Now we have two equations:
615 = 7.5a + 52.5d ---> (Equation 1)
6d = 5a ---> (Equation 2)
We can solve these equations simultaneously to find the values of a and d.
Step 3: Solving the equations:
Using substitution, we can substitute 5a from Equation 2 into Equation 1.
615 = 7.5a + 52.5d
615 = 7.5(6d) + 52.5d
615 = 45d + 52.5d
615 = 97.5d
d = 615/97.5
d = 6.3
Substituting the value of d back into Equation 2 to find a:
6d = 5a
6(6.3) = 5a
37.8 = 5a
a = 37.8/5
a = 7.56
Now that we have the values of a and d, we can find the first three terms of the arithmetic sequence.
The first term (a) = 7.56
The second term (a + d) = 7.56 + 6.3 = 13.86
The third term (a + 2d) = 7.56 + 2(6.3) = 20.16
Therefore, the first three terms of the arithmetic sequence are 7.56, 13.86, and 20.16.
To find the first three terms of an arithmetic sequence, we need to find the common difference (d) and then use it to determine the values of the first three terms.
First, let's find the common difference (d). We can use the formula for the sum of an arithmetic series:
Sn = (n/2)(2a + (n-1)d)
In this formula, Sn represents the sum of the first n terms, a represents the first term, and d represents the common difference.
We are given that the sum of the first 15 terms is 615:
615 = (15/2)(2a + 14d)
Now, let's find another equation by using the given information that the 13th term is six times the second term:
a + 12d = 6(a + d)
Expand and simplify:
a + 12d = 6a + 6d
6d - 6a = 0
d - a = 0
Now, we have a system of two equations:
615 = (15/2)(2a + 14d)
d - a = 0
We can solve this system of equations to find the values of a and d.
First, let's solve d - a = 0 for a in terms of d:
a = d
Now, substitute this into the first equation:
615 = (15/2)(2a + 14d)
615 = (15/2)(2d + 14d)
615 = (15/2)(16d)
41 = 16d
d = 41/16
Now that we know d, we can find a by substituting it back into the equation d = a:
41/16 = a
Therefore, the first term (a) is 41/16 and the common difference (d) is 41/16.
To find the first three terms, we can now substitute these values into the formula for the arithmetic sequence:
First term (a) = 41/16
Second term (a + d) = 41/16 + 41/16 = 82/16
Third term (a + 2d) = 41/16 + 2(41/16) = 41/16 + 82/16 = 123/16
So, the first three terms of the arithmetic sequence are 41/16, 82/16, 123/16.