the 9th term of an arithmetic sequence is 29. the 16th term is 50. what is the 20th term ?
well, the difference d = (50-29)/(16-9) = 3
T20 = T16 + 4d
...
To find the 20th term of an arithmetic sequence, we can use the formula:
an = a1 + (n - 1)d
where:
an represents the nth term,
a1 represents the first term of the sequence, and
d represents the common difference.
We are given that the 9th term of the sequence is 29. Let's use this information to find the value of a9. We can plug the values into the equation:
29 = a1 + (9 - 1)d
Simplifying this equation, we get:
29 = a1 + 8d ...(equation 1)
Next, we are given that the 16th term of the sequence is 50. Let's use this information to find the value of a16. We can plug the values into the equation:
50 = a1 + (16 - 1)d
Simplifying this equation, we get:
50 = a1 + 15d ...(equation 2)
Now, we have two equations - equation 1 and equation 2 - with the same a1 variable and two different sets of values for n and d. We can solve these equations simultaneously to find the values of a1 and d.
Subtracting equation 1 from equation 2, we can eliminate the a1 term:
50 - 29 = (a1 + 15d) - (a1 + 8d)
Simplifying this equation, we get:
21 = 7d
Dividing both sides of the equation by 7, we can isolate the value of d:
d = 21 / 7
d = 3
Substituting the value of d back into equation 1, we can solve for a1:
29 = a1 + 8(3)
29 = a1 + 24
Subtracting 24 from both sides of the equation, we get:
29 - 24 = a1
5 = a1
Now that we have the values of a1 and d, we can find the 20th term of the arithmetic sequence using the formula:
a20 = a1 + (20 - 1)d
Plugging in the values, we get:
a20 = 5 + (20 - 1)3
Simplifying this equation, we find:
a20 = 5 + 19(3)
a20 = 5 + 57
a20 = 62
Therefore, the 20th term of the arithmetic sequence is 62.