The fith term of an arithmetic sequence is 23 and the 21th term is 72 Determine the first three terms of the sequence and the nth term
Tn=a+(5-1)d=23 so 23=a+4d T12=a+(12-1)d so 72=a+11d
To find the first three terms of the arithmetic sequence, we can use the formulas for arithmetic sequences.
The general form of an arithmetic sequence is given by the formula:
an = a1 + (n - 1)d
where an is the nth term, a1 is the first term, n is the position of the term in the sequence, and d is the common difference between consecutive terms.
Given that the fifth term (a5) is 23, we can substitute the values into the formula:
23 = a1 + (5 - 1)d
Simplifying, we have:
23 = a1 + 4d ...........(1)
Similarly, given that the 21st term (a21) is 72, we can substitute the values into the formula:
72 = a1 + (21 - 1)d
Simplifying this equation, we have:
72 = a1 + 20d ...........(2)
Now we have a system of two equations with two variables (a1 and d). We can solve these equations to find the values of a1 and d.
Subtracting equation (1) from equation (2), we get:
72 - 23 = (a1 + 20d) - (a1 + 4d)
49 = 16d
Dividing both sides by 16, we get:
d = 49/16
Substituting this value of d back into equation (1), we can find the value of a1:
23 = a1 + 4(49/16)
23 = a1 + 196/16
23 = a1 + 49/4
Now, let's find the common difference (d) and the first term (a1) of the sequence:
d = 49/16
a1 = 23 - 49/4
To find the first term, a1, subtract 49/4 from 23:
a1 = (92 - 49)/4
a1 = 43/4
Therefore, the first term (a1) is 43/4 and the common difference (d) is 49/16.
To find the nth term, we can use the formula:
an = a1 + (n - 1)d
Substituting the values we found:
an = (43/4) + (n - 1)(49/16)
Simplifying this expression gives us the formula for the nth term of the sequence.
So, the first three terms of the sequence are:
a1 = 43/4
a2 = 43/4 + (2 - 1)(49/16)
a3 = 43/4 + (3 - 1)(49/16)
And the formula for the nth term of the sequence is:
an = (43/4) + (n - 1)(49/16)