Find the 265th term of an arithmetic sequence whose second term is 5/8 and whose 9th term is 71/24 pls help asap
a+d = 5/8
a + 8d = 71/24
subtract
7d = 7/3
d = 1/3
then in a+d = 5/8
a = 5/8 - 1/3 = 7/24
term 265 = a + 264d
=7/24 + 264/3 = 2119/24
To find the 265th term of an arithmetic sequence, we need to determine the common difference (d) of the sequence. Once we have the common difference, we can use the formula for the nth term of an arithmetic sequence to find the desired term.
Let's begin by finding the common difference (d):
We are given the second term, which is 5/8, and the ninth term, which is 71/24.
Using the formula for the nth term of an arithmetic sequence, we can write the equations:
a + (n - 1)d = t₁, where a is the first term (unknown), n is the position of the term, and t₁ is the given term.
For the second term, substituting n = 2 and t₁ = 5/8, we get:
a + (2 - 1)d = 5/8
a + d = 5/8 (Equation 1)
Similarly, for the ninth term, substituting n = 9 and t₁ = 71/24, we get:
a + (9 - 1)d = 71/24
a + 8d = 71/24 (Equation 2)
We now have a system of two equations (Equation 1 and Equation 2) with two variables (a and d). We can solve this system to find the values of a and d.
To solve this system, we'll multiply Equation 1 by 8 and Equation 2 by 3 to eliminate the fractions:
8(a + d) = 8(5/8)
24(a + 8d) = 24(71/24)
This simplifies to:
8a + 8d = 5
24a + 192d = 71
Let's use elimination or substitution to solve this system of equations:
Multiply Equation 1 by 3:
3(a + d) = 3(5/8)
3a + 3d = 15/8 (Equation 3)
Subtract Equation 3 from Equation 2:
(24a + 192d) - (3a + 3d) = (71) - (15/8)
24a + 192d - 3a - 3d = 568/8 - 15/8
21a + 189d = 553/8
Now, we have a new equation:
21a + 189d = 553/8 (Equation 4)
Let's solve this new equation (Equation 4) with Equation 3 to eliminate one variable:
Multiply Equation 3 by 7:
7(3a + 3d) = 7(15/8)
21a + 21d = 105/8 (Equation 5)
Subtract Equation 5 from Equation 4:
(21a + 189d) - (21a + 21d) = (553/8) - (105/8)
21a + 189d - 21a - 21d = 448/8
168d = 448/8
d = (448/8) / 168
d = (448/8) * (1/168)
d = 28/168
d = 1/6
Now that we have found the common difference (d = 1/6), we can find the first term (a) using Equation 1:
a + d = 5/8
a + 1/6 = 5/8
a = 5/8 - 1/6
a = (15/24) - (4/24)
a = 11/24
Thus, the first term (a) is 11/24 and the common difference (d) is 1/6.
Now, we can use the formula for the nth term of an arithmetic sequence to find the 265th term:
tₙ = a + (n - 1)d
Substituting the known values:
tₙ = (11/24) + (265 - 1)(1/6)
tₙ = (11/24) + (264/6)
tₙ = (11/24) + (44/24)
tₙ = 55/24
Therefore, the 265th term of the arithmetic sequence is 55/24.