in an arithmetic sequence 17th term is 9 times the first three terms and the 9th term is 6 less than 3 times the 1st term. Find the first three terms
a+16d = 9(a + a+d + a+2d)
a+16d = 27a + 27d
-11d = 26a
a+8d = 3a - 6
8d+6 = 2a or
a = 4d+3 <----- sub that into the first equation
-11d = 26(4d+3)
-11d = 104d + 78
-115d = 78
d = -78/115
a = 4(-78/115) + 3 = 33/115
the first 3 terms are : 33/115 , -9/23, -123/115
check:
the sum of the first 3 is -27/23
9 times that sum = -243/23
17th term = -243/23 , check!
9th term = -591/115
3 times the first - 6 = -591/115 , check
I was expecting nicer numbers, but my answer works.
To find the first three terms of the arithmetic sequence, let's first find an equation based on the given information.
Let the common difference of the arithmetic sequence be "d", and let the first term be "a".
From the given information, we know that:
The 17th term is 9 times the sum of the first three terms.
So, the 17th term can be expressed as:
a + 16d = 9(a + (a+d) + (a+2d))
Simplifying the equation:
a + 16d = 9(3a + 3d)
a + 16d = 27a + 27d
-26a + 11d = 0 ----(1)
We are also given that:
The 9th term is 6 less than 3 times the first term.
So, the 9th term can be expressed as:
a + 8d = 3a - 6
Simplifying the equation:
2a - 8d = 6 ----(2)
Now we have a system of equations (1) and (2) with two variables. We can solve it using the elimination method or substitution method.
Let's use the elimination method to solve the system of equations:
Multiply equation (1) by 8 and add it to equation (2):
-208a + 88d = 0
2a - 8d = 6
Adding these equations together eliminates the "d" variable:
-206a = 6
Divide both sides by -206:
a = -6/206
Simplify:
a = -3/103
Now that we have found the value of the first term (a), we can substitute it into one of the equations to find the common difference (d). Let's use equation (1):
-26a + 11d = 0
-26(-3/103) + 11d = 0
78/103 + 11d = 0
11d = -78/103
Divide both sides by 11:
d = -78/1133
Therefore, the first three terms of the arithmetic sequence are:
a = -3/103
d = -78/1133
To find the second and third terms, we can use the formulas:
2nd term = 1st term + common difference
3rd term = 2nd term + common difference
Substituting the values we found:
2nd term = (-3/103) + (-78/1133)
3rd term = (-3/103) + (-78/1133) + (-78/1133)
Performing the addition and simplification:
2nd term ≈ -0.032
3rd term ≈ -0.036
Therefore, the first three terms of the arithmetic sequence are approximately:
1st term ≈ -3/103
2nd term ≈ -0.032
3rd term ≈ -0.036