The sum of the first 9 terms of an arithmetic series is 144, and the sum of the first 14 terms is 329. Find the first 4 terms of the series.
Fhhh
To find the first 4 terms of the arithmetic series, let's first find the common difference (d) and the first term (a1).
We know that the sum of the first 9 terms of the series is 144. Using the formula for the sum of an arithmetic series, we can write this equation:
S9 = (9/2)(2a1 + (9 - 1)d) = 144
Simplifying the equation, we get:
(9/2)(2a1 + 8d) = 144
18a1 + 72d = 288
Now, let's use the second piece of information. The sum of the first 14 terms of the series is 329:
S14 = (14/2)(2a1 + (14 - 1)d) = 329
Simplifying the equation, we get:
(14/2)(2a1 + 13d) = 329
14a1 + 91d = 658
Now, we have a system of two equations:
18a1 + 72d = 288 (Equation 1)
14a1 + 91d = 658 (Equation 2)
We can solve this system to find the values of a1 and d.
First, let's multiply Equation 1 by 7 and Equation 2 by 9 to eliminate the variable 'a1':
7(18a1 + 72d) = 7(288) -> 126a1 + 504d = 2016 (Equation 3)
9(14a1 + 91d) = 9(658) -> 126a1 + 819d = 5922 (Equation 4)
Now, subtract Equation 3 from Equation 4 to get:
(126a1 + 819d) - (126a1 + 504d) = 5922 - 2016
315d = 3906
Dividing both sides by 315, we find that d = 12.
Now, substitute the value of d into either Equation 1 or Equation 2 to solve for a1:
18a1 + 72(12) = 288
18a1 + 864 = 288
18a1 = -576
Dividing both sides by 18, we find that a1 = -32.
So, the common difference (d) is 12 and the first term (a1) is -32.
Now, we can find the first 4 terms of the series:
a1 = -32
a2 = a1 + d = -32 + 12 = -20
a3 = a2 + d = -20 + 12 = -8
a4 = a3 + d = -8 + 12 = 4
Therefore, the first 4 terms of the arithmetic series are -32, -20, -8, and 4.
To find the first 4 terms of the arithmetic series, let's break down the problem step by step:
Step 1: Find the common difference (d)
An arithmetic series is a sequence of numbers where the difference between consecutive terms is constant. Let's find the common difference (d) first.
For the sum of the first 9 terms, we have the equation:
9/2 * [2a + (9-1)d] = 144
Simplifying this equation, we get:
9/2 * (2a + 8d) = 144
18a + 72d = 144
18a + 72d - 144 = 0
2a + 8d - 16 = 0
a + 4d - 8 = 0
Similarly, for the sum of the first 14 terms, we have the equation:
14/2 * [2a + (14-1)d] = 329
Simplifying this equation, we get:
14/2 * (2a + 13d) = 329
7a + 91d = 329
7a + 91d - 329 = 0
a + 13d - 47 = 0
Step 2: Solve the system of equations
Now, we have two equations with two unknowns (a and d). We can solve this system of equations to find the values of a and d.
Using the elimination method, we can subtract the second equation from the first equation:
(a + 4d - 8) - (a + 13d - 47) = 0
a - a + 4d - 13d - 8 + 47 = 0
-9d + 39 = 0
-9d = -39
d = -39 / -9
d = 4.333
Step 3: Find the first term (a)
Substituting the value of d into one of the original equations, let's solve for a:
a + 4(4.333) - 8 = 0
a + 17.332 - 8 = 0
a + 9.332 = 0
a = -9.332
Step 4: Find the first 4 terms
Now that we have the values of a and d, we can find the first 4 terms of the arithmetic series.
The first term (a) is -9.332, and the common difference (d) is 4.333.
The first term of the series is a = -9.332.
The second term is a + d = -9.332 + 4.333 = -4.999.
The third term is a + 2d = -9.332 + 2(4.333) = 1.666.
The fourth term is a + 3d = -9.332 + 3(4.333) = 7.999.
Therefore, the first 4 terms of the arithmetic series are -9.332, -4.999, 1.666, and 7.999.
aaarrrrggh too many of these, use:
http://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html