The second and ninth terms of an arithmetic sequence are 2 and 30 respectively. What is the fiftieth term.
t2 = a + d = 2
t9 = a + 8d = 30
subtract them
7x = 28
d = 4
in a+d=2
a + 4 = 2
a = -2
t15 = a + 14d = -2 + 14(4) = 54
It is the fiftieth term not the fifteenth so would it be t50=a+49d
To find the fiftieth term of an arithmetic sequence, we need to first find the common difference. The common difference (d) is the constant value added or subtracted to each term to get to the next term.
Given that the second term (a2) is 2 and the ninth term (a9) is 30, we can use these values to find the common difference.
The formula to find the nth term (an) of an arithmetic sequence is:
an = a1 + (n-1)d
In this case, we know the values for a2 and a9. We can substitute these values into the formula to get two equations:
For a2:
2 = a1 + (2-1)d
2 = a1 + d -- Equation 1
For a9:
30 = a1 + (9-1)d
30 = a1 + 8d -- Equation 2
Now, we have two equations with two variables (a1 and d). We can solve this system of equations to find the values of a1 and d.
Subtract Equation 1 from Equation 2:
28 = 7d
Divide by 7:
d = 4
Now that we have the common difference (d) value, we can find the first term (a1) using Equation 1:
2 = a1 + 4
a1 = -2
Now, we know the first term (a1 = -2) and the common difference (d = 4) of the arithmetic sequence. We can use the formula to find the fiftieth term (a50):
a50 = a1 + (50-1)d
a50 = -2 + (49)(4)
a50 = -2 + 196
a50 = 194
Therefore, the fiftieth term of the arithmetic sequence is 194.