2
The 11t ^ h term of an arithmetic sequence is 57 and the sum of the first and fourth terms is 29.
2.3 Is 100 a term in the sequence? Justify your answer.
To solve this problem, we need to find the common difference (d) of the arithmetic sequence.
Let's call the first term a1 and the common difference d.
The 11th term of the arithmetic sequence can be found using the formula:
a11 = a1 + (11 - 1) * d
Since a11 is given as 57, we can substitute it into the formula:
57 = a1 + 10d ----(1)
The sum of the first and fourth terms can be found using the formula:
a1 + a4 = 29
Substituting the formula for the nth term:
a1 + (1 - 1) * d + a1 + (4 - 1) * d = 29
2a1 + 3d = 29 ----(2)
Now we have a system of two equations (1) and (2) with two variables (a1 and d).
We can solve this system of equations to find the values of a1 and d.
From equation (2), we can solve for a1 in terms of d:
2a1 = 29 - 3d
a1 = (29 - 3d)/2 ----(3)
Substituting equation (3) into equation (1):
57 = (29 - 3d)/2 + 10d
Multiplying both sides of the equation by 2 to eliminate the fraction:
114 = 29 - 3d + 20d
Combining like terms:
114 = 29 + 17d
85 = 17d
d = 5
Now we can substitute this value of d into equation (3) to find a1:
a1 = (29 - 3*5)/2
a1 = 29 - 15/2
a1 = 29 - 7.5
a1 = 21.5
Therefore, the first term (a1) is 21.5 and the common difference (d) is 5.
To determine whether 100 is a term in the sequence, we can use the formula for the nth term:
an = a1 + (n - 1) * d
Substituting the values we found:
100 = 21.5 + (n - 1) * 5
Simplifying:
100 = 21.5 + 5n - 5
100 - 21.5 + 5 = 5n
83.5 = 5n
n = 16.7
Since n is not a whole number, 100 is not a term in the arithmetic sequence.