Write an explicit formula for a_na
n
,the n^{\text{th}}n
thterm of the sequence 5, 15, 25, ...5,15,25,....
Write an explicit formula for a_na
n
, the n^{\text{th}}n
th
term of the sequence 1, 5, 25, ..
The given sequence is an arithmetic sequence with a common difference of 10.
To find the explicit formula for the nth term (a_n), we can use the formula for nth term of an arithmetic sequence:
a_n = a_1 + (n - 1)d
Where a_1 is the first term of the sequence, d is the common difference, and n is the position of the term we want to find.
In this case, the first term of the sequence (a_1) is 5 and the common difference (d) is 10.
Plugging these values into the formula, we get:
a_n = 5 + (n - 1) * 10
Simplifying further:
a_n = 5 + 10n - 10
a_n = 10n - 5
So, the explicit formula for the nth term of the sequence 5, 15, 25, ... is a_n = 10n - 5.
To find the explicit formula for a_na
, the n
th term of the sequence 5, 15, 25, ..., we need to observe the pattern and derive a general formula.
From the given sequence, we can see that each term is obtained by adding 10 to the previous term. So, we can express this relationship as:
a_n = a_{n-1} + 10
To find the first term of the sequence, we can substitute n = 1 into the formula. Since the first term is given as 5, we have:
a_1 = a_{1-1} + 10
5 = a_0 + 10
Next, we solve for a_0. Subtracting 10 from both sides gives us:
-5 = a_0
So, we have found the value of a_0 as -5.
Now, we can rewrite the formula for a_n with the initial term:
a_n = -5 + 10n
Therefore, the explicit formula for a_na
, the n
th term of the given sequence, is:
a_n = -5 + 10n
if a = 15 and d = 10
Tn = a + d(n-1)