The 10th term of an arithmetic sequence 17 and the16th is 44,find the first three terms in the sequence
a+9d = 17
a+15d = 44
subtract them
6d = 27
d = 4.5
then a = 17-9(4.5) = -23.5
sequence starts as
-23.5 , -19 , -14.5 ...
To find the first three terms of an arithmetic sequence, we need to determine the common difference (d) between consecutive terms.
The formula for the nth term of an arithmetic sequence is given by:
An = A1 + (n - 1)d
Given that the 10th term (A10) is 17 and the 16th term (A16) is 44, we can use these values to set up two equations and solve for the common difference.
Equation 1: A10 = A1 + 9d
17 = A1 + 9d
Equation 2: A16 = A1 + 15d
44 = A1 + 15d
To eliminate A1, we can subtract Equation 1 from Equation 2:
A16 - A10 = (A1 + 15d) - (A1 + 9d)
44 - 17 = 15d - 9d
27 = 6d
Now, we can solve for d:
6d = 27
d = 27/6
d = 4.5
We have found the common difference to be 4.5.
To find the first term (A1), we can substitute this value of d into either Equation 1 or 2:
17 = A1 + 9(4.5)
17 = A1 + 40.5
A1 = 17 - 40.5
A1 = -23.5
Thus, the first term (A1) is -23.5.
Finally, to find the first three terms, we can use the equation:
A1 = -23.5
A2 = A1 + d
A3 = A1 + 2d
Substituting the values:
A1 = -23.5
A2 = -23.5 + 4.5 = -19
A3 = -23.5 + 2(4.5) = -23.5 + 9 = -14.5
Therefore, the first three terms in the arithmetic sequence are:
A1 = -23.5
A2 = -19
A3 = -14.5