THE FIFTH TERM OF AN ARITHMETIC SEQUENCE IS 23 AND THE 12TH TERM IS 72.AND THEY SAY DETERMINE THE FIRST THREE TERMS OF THE SEQUENCE AND NTH TERM
you have to know the formula for the general term of an AS.
term(n) = a + (n-1)d
term5 = a+ 4d = 23
term12 = a + 11d = 72
subtract them
7d = 49
d = 7
sub into a+4d = 23
a + 28 = 23
a = -5
first three terms are -5 , 2, and 9
How do i get to the answers as shown above
To determine the first three terms of the arithmetic sequence and the nth term, we can use the formula for the nth term of an arithmetic sequence.
The general formula for the nth term of an arithmetic sequence is:
an = a1 + (n - 1)d
Where "an" represents the nth term of the sequence, "a1" is the first term, "n" is the position of the term in the sequence, and "d" is the common difference between consecutive terms.
In this case, we have two pieces of information: the 5th term (a5 = 23) and the 12th term (a12 = 72). We can use these values to create two equations and solve for the first term (a1) and the common difference (d).
Equation 1: a5 = a1 + 4d
Equation 2: a12 = a1 + 11d
Substituting the given values, we get:
23 = a1 + 4d (Equation 1)
72 = a1 + 11d (Equation 2)
Now, we can solve these two equations simultaneously to find the values of a1 and d.
Subtracting Equation 1 from Equation 2, we eliminate a1 and obtain:
49 = 7d
Dividing both sides by 7, we get:
d = 7
Substituting the value of d back into Equation 1 or Equation 2, we can solve for a1.
Using Equation 1: 23 = a1 + 4(7)
23 = a1 + 28
a1 = 23 - 28
a1 = -5
Therefore, the first term (a1) of the arithmetic sequence is -5, and the common difference (d) is 7.
To find the first three terms of the sequence, we substitute the values of a1 and d into the formula for the nth term.
a1 = -5,
d = 7
The first three terms of the sequence can be calculated as follows:
a1 = -5
a2 = a1 + d = -5 + 7 = 2
a3 = a1 + 2d = -5 + 2(7) = 9
So, the first three terms of the arithmetic sequence are:
-5, 2, 9
To find the nth term of the arithmetic sequence, we can use the formula:
an = a1 + (n-1)d
Substituting the values we found earlier:
an = -5 + (n-1)(7)
an = -5 + 7n - 7
an = 7n - 12
Therefore, the nth term (an) of the arithmetic sequence is given by the formula 7n - 12.
Well, let's see if we can solve this arithmetic puzzle together. To find the first three terms of the sequence, we can use the formula for the nth term of an arithmetic sequence:
nth term = a + (n-1)d
Where "a" is the first term, "n" is the position of the term in the sequence, and "d" is the common difference between the terms.
Given that the fifth term is 23, we can plug in the values and solve for "d":
23 = a + (5-1)d
23 = a + 4d
We also know that the 12th term is 72:
72 = a + (12-1)d
72 = a + 11d
Now we have a system of linear equations:
23 = a + 4d - (Equation 1)
72 = a + 11d - (Equation 2)
Let's solve this clownishly, shall we?
Subtracting Equation 1 from Equation 2:
49 = 7d
Dividing both sides by 7:
d = 7
Now that we have the value of "d", we can substitute it back into Equation 1:
23 = a + 4(7)
23 = a + 28
-5 = a
Therefore, the first term (a) is -5, and the common difference (d) is 7.
To find the nth term, we can use the same formula mentioned earlier:
nth term = a + (n-1)d
So, substituting the known values:
nth term = -5 + (n-1)7
And there you have it! The first three terms of the sequence are -5, 2, and 9. And the nth term of the sequence is -5 + (n-1)7.
Now that we've solved it, I hope this math problem did not take... too much of your "dough" time!