IN AN ARITHMETICS SEQUENCE THE SEVENTH TERM IS 12 AND THE 21ST TERM IS 68. CALCULATE THE 68TH TERM.
a+6d=12.......(1)
a+20d=68......(2)
solve by elimination
a is gone
14d=56
d=4
a+6*4=12
a=12-24
a=-12
68 term=a+67d=?
that reminds me the equation i use to solve is
nth=a+(n-1)d
To find the 68th term in an arithmetic sequence, we need to determine the common difference.
First, let's find the common difference (d) using the given terms.
The formula to find the nth term of an arithmetic sequence is:
Tn = a + (n - 1) * d
Where:
Tn = nth term
a = first term
n = number of terms
d = common difference
Given:
Seventh term (T7) = 12
Twenty-first term (T21) = 68
Using the formula, we have:
T7 = a + (7 - 1) * d = 12
T21 = a + (21 - 1) * d = 68
We can use these equations to solve for the common difference (d):
a + (7 - 1) * d = 12
a + 6d = 12 --> Equation 1
a + (21 - 1) * d = 68
a + 20d = 68 --> Equation 2
To isolate 'a', we need to eliminate 'a' from these two equations.
Multiplying Equation 1 by 20, and Equation 2 by 6 will give us two equations to eliminate 'a' from:
20a + 120d = 240
6a + 120d = 408
Subtracting these equations will eliminate 'a':
(20a + 120d) - (6a + 120d) = 240 - 408
14a = -168
Dividing by 14 on both sides, we can solve for 'a':
14a/14 = -168/14
a = -12
Now that we have the value of 'a', we can substitute it back into Equation 1 or 2 to find 'd'. Let's use Equation 1:
a + 6d = 12
Substituting -12 for 'a':
-12 + 6d = 12
Adding 12 to both sides:
6d = 24
Dividing by 6 on both sides, we can solve for 'd':
6d/6 = 24/6
d = 4
Now that we have the common difference 'd' as 4 and the first term 'a' as -12, we can use the formula to find the 68th term (T68):
T68 = a + (68 - 1) * d
= -12 + 67 * 4
= -12 + 268
= 256
Therefore, the 68th term of the arithmetic sequence is 256.
To calculate the 68th term of an arithmetic sequence, you first need to determine the common difference (d) between consecutive terms. The common difference remains constant throughout the sequence.
Given that the 7th term is 12 and the 21st term is 68, we can use these two terms to find the common difference.
Formula:
n-th term of an arithmetic sequence = a + (n - 1) * d
Here, a represents the first term, n represents the position of the term in the sequence, and d represents the common difference.
Step 1: Finding the common difference (d)
Using the formula, we can substitute the values of the 7th and 21st terms to find the common difference.
12 = a + (7 - 1) * d
68 = a + (21 - 1) * d
Simplifying these equations gives us:
12 = a + 6d
68 = a + 20d
Step 2: Solving for a and d
Now, we can solve the above two equations as a system of linear equations to find the values of a and d.
Let's multiply the first equation by 10 and the second equation by 3 to make the coefficients of 'd' the same:
120 = 10a + 60d
204 = 3a + 60d
Subtracting the second equation from the first gives:
120 - 204 = 10a - 3a
-84 = 7a
Dividing both sides by 7, we find:
a = -12
Now, substitute the value of a (first term) into either of the initial equations (let's use 12 = a + 6d) and solve for d:
12 = -12 + 6d
24 = 6d
d = 4
Step 3: Calculating the 68th term
Now we know that the first term (a) is -12 and the common difference (d) is 4. We can use the formula for the n-th term of an arithmetic sequence to calculate the 68th term.
n = 68
a = -12
d = 4
68th term = -12 + (68 - 1) * 4
68th term = -12 + 67 * 4
68th term = -12 + 268
68th term = 256
Therefore, the 68th term of the arithmetic sequence is 256.