The second term of an arithmetic sequence is 24 and the fifth term is 3.Find the first term and the common difference.
To find the first term and the common difference of an arithmetic sequence, we can use the formulas for the nth term of an arithmetic sequence, given as:
nth term = a + (n - 1) * d,
where "a" is the first term, "n" is the number of the term in the sequence, and "d" is the common difference.
Given that the second term (n = 2) of the sequence is 24 and the fifth term (n = 5) is 3, we can now set up two equations to solve for the unknowns (a and d).
Using the formula for the second term:
24 = a + (2 - 1) * d
24 = a + d
Using the formula for the fifth term:
3 = a + (5 - 1) * d
3 = a + 4d
Now we have a system of two equations:
1) a + d = 24
2) a + 4d = 3
To solve this system, we can use the method of substitution or elimination. Let's use the method of elimination.
Multiply equation 1 by 4 to eliminate "a":
4(a + d) = 4 * 24
4a + 4d = 96
Subtract equation 2 from the modified equation 1:
(4a + 4d) - (a + 4d) = 96 - 3
4a + 4d - a - 4d = 93
3a = 93
a = 93 / 3
a = 31
Now we have found the value of the first term, which is a = 31.
Substitute the value of a into equation 1 to solve for d:
31 + d = 24
d = 24 - 31
d = -7
Therefore, the first term of the arithmetic sequence is 31, and the common difference is -7.
Use your formulas.
a+d = 24
a+4d = 3
subtract them:
3c = -21
d = -7
in a+d=24
a=7=24
a = 31
first term is 31, common difference is -7