If the 16th term of an arithmetic sequence is three times the fourth term, find the ratio of the 23rd term to the third term.
To solve this problem, we can use the formula for the nth term of an arithmetic sequence:
a[n] = a[1] + (n - 1)d
where a[n] is the nth term, a[1] is the first term, n is the term number, and d is the common difference.
Let's denote the fourth term as a[4] and the 16th term as a[16]. We are given that a[16] is three times a[4]. Mathematically, it can be expressed as:
a[16] = 3a[4]
We can also find the difference between the terms, which is the common difference (d). Using the formula above, we can compare the terms:
a[4] = a[1] + (4 - 1)d
a[16] = a[1] + (16 - 1)d
Since the common difference is the same for both terms, we can subtract the first equation from the second equation to eliminate a[1]:
a[16] - a[4] = (a[1] + (16 - 1)d) - (a[1] + (4 - 1)d)
Simplifying the equation, we get:
3a[4] - a[4] = 15d
2a[4] = 15d
Now we need to find the ratio of the 23rd term (a[23]) to the third term (a[3]). Using the formula for the nth term, we can write the expressions for these terms:
a[23] = a[1] + (23 - 1)d
a[3] = a[1] + (3 - 1)d
To find the ratio, we divide the expression for the 23rd term by the expression for the third term:
a[23] / a[3] = (a[1] + (23 - 1)d) / (a[1] + (3 - 1)d)
Now we need to eliminate a[1] using the information we have. We can consider the difference between the 16th term and the 4th term:
3a[4] = a[16]
Solving for a[4], we get:
a[4] = a[16] / 3
Substituting this expression into the ratio equation, we have:
a[23] / a[3] = ((a[16] / 3) + (23 - 1)d) / ((a[16] / 3) + (3 - 1)d)
Simplifying further, we have:
a[23] / a[3] = (a[16] + 6d) / (a[16] + 2d)
Since we have the equation 2a[4] = 15d, we can express d in terms of a[4]:
15d = 2a[4]
d = 2a[4] / 15
Substituting this expression into the ratio equation, we get:
a[23] / a[3] = (a[16] + 6 * (2a[4] / 15)) / (a[16] + 2 * (2a[4] / 15))
Simplifying further, we have:
a[23] / a[3] = (a[16] + 12a[4] / 15) / (a[16] + 4a[4] / 15)
This is the expression for the ratio of the 23rd term to the third term in the arithmetic sequence.
To find the ratio of the 23rd term to the third term, we first need to find the values of the 23rd term and the third term separately.
Let's denote the first term of the arithmetic sequence as 'a' and the common difference as 'd'.
Since the 16th term is three times the fourth term, we can write the equation:
a + 15d = 3(a + 3d)
To simplify this equation, let's expand and rearrange it:
a + 15d = 3a + 9d
6d = 2a
3d = a
Now that we have the relationship between 'a' and 'd', we can express the 23rd term and the third term in terms of 'a' and 'd':
The 23rd term (T23) = a + 22d
The third term (T3) = a + 2d
Using the relationship we found earlier, we substitute '3d' for 'a':
T23 = 3d + 22d = 25d
T3 = 3d + 2d = 5d
Finally, we can calculate the ratio of the 23rd term to the third term:
T23/T3 = (25d)/(5d) = 25/5 = 5
Therefore, the ratio of the 23rd term to the third term is 5.
translate from English to Math
"If the 16th term of an arithmetic sequence is three times the fourth term"
--->a+15d = 3(a+3d)
a+15d = 3a+3d
12d = 2a
a = 6d
term 23 = a+22d = 6d+22d = 28d
term 3 = a + 2d = 6d + 2d = 8d
term23 : term3 = 28d : 8d
= 28 : 8
= 7 : 2