The sum of the first 16 terms of an arithmetic sequence is 23 times the sum of the first 4 terms. Find the 10th term of the sequence if the first term is 2
To find the 10th term of an arithmetic sequence, we need to determine the common difference and then use the formula for the nth term of an arithmetic sequence.
Given that the first term is 2, we can assume the arithmetic sequence is as follows:
2, __, __, __, __, __, __, __, __, __, __, __, __, __, __, __, __
Let's use the given information to find the common difference and then proceed to find the 10th term.
The sum of the first n terms of an arithmetic sequence can be found using the formula:
Sn = (n/2)(2a + (n-1)d)
where Sn is the sum of the first n terms, a is the first term, n is the number of terms, and d is the common difference.
From the given information, we know that the sum of the first 16 terms of the sequence is 23 times the sum of the first 4 terms. Mathematically, this can be written as:
(16/2)(2*2 + (16-1)d) = 23[(4/2)(2*2 + (4-1)d)]
Simplifying the equation, we have:
(8)(4 + 15d) = 23(2 + 6d)
Expanding both sides of the equation:
32 + 120d = 46 + 138d
Moving all the terms involving d to one side of the equation:
18d = 14
Solving for d:
d = 14/18
d = 7/9
Now that we have the common difference, we can find the 10th term using the formula for the nth term of an arithmetic sequence:
an = a + (n-1)d
where an is the nth term, a is the first term, n is the position of the term, and d is the common difference.
Applying the formula:
a10 = 2 + (10-1)(7/9)
a10 = 2 + 9(7/9)
a10 = 2 + 7
a10 = 9
Therefore, the 10th term of the arithmetic sequence is 9.