the 3rd of an Arithmetic sequence is 8 of the 16th term is 47. Find a subed 1 and the common difference. Construct the sequence.
a3 and a16 are 13 terms apart. So,
13d = 47-8 = 39
d=3
a+2d=8, so a=2
Now you know a and d.
Sagot lang po
To find the first term (a sub 1) and the common difference (d) of an arithmetic sequence, we can use the given information.
Let's start by finding the first term (a sub 1):
We're given that the 3rd term of the sequence is 8. In an arithmetic sequence, the formula for finding the nth term is given by:
a sub n = a sub 1 + (n - 1) * d
Substituting n = 3 and a sub n = 8 into the formula, we get:
8 = a sub 1 + (3 - 1) * d
8 = a sub 1 + 2d
Now, let's find the 16th term:
We're given that the 16th term of the sequence is 47. Applying the same formula, we get:
47 = a sub 1 + (16 - 1) * d
47 = a sub 1 + 15d
Now, we have two equations:
1) 8 = a sub 1 + 2d
2) 47 = a sub 1 + 15d
To solve these equations, we can use elimination or substitution method. For simplicity, let's solve using substitution.
Rearrange equation 1) to isolate a sub 1:
a sub 1 = 8 - 2d
Substitute this value into equation 2):
47 = (8 - 2d) + 15d
Now solve for d:
47 = 8 + 13d
39 = 13d
d = 3
Substitute this value back into equation 1) to find a sub 1:
a sub 1 = 8 - 2 * 3 = 8 - 6 = 2
So, the common difference (d) is 3 and the first term (a sub 1) is 2.
To construct the sequence, we now have the values of a sub 1 and d. Using the arithmetic sequence formula, we can find each term in the sequence.
The formula for the nth term of an arithmetic sequence is:
a sub n = a sub 1 + (n - 1) * d
Plugging in the values, we have:
a sub n = 2 + (n - 1) * 3
Now, we can calculate the terms of the sequence for n = 1, 2, 3, ... until 16.
a sub 1 = 2 + (1 - 1) * 3 = 2
a sub 2 = 2 + (2 - 1) * 3 = 5
a sub 3 = 2 + (3 - 1) * 3 = 8
...
Continuing the pattern, we can calculate all 16 terms of the sequence in this way.