For the sequence an = 4n + 5:
find: the first five terms:
find the sum of the first 25 terms:
is it a arithmetic sequence? If so how can you tell, if not, why not?
To find the first five terms of the sequence an = 4n + 5, plug in values of n and calculate the corresponding term:
For n = 1: a1 = 4(1) + 5 = 9
For n = 2: a2 = 4(2) + 5 = 13
For n = 3: a3 = 4(3) + 5 = 17
For n = 4: a4 = 4(4) + 5 = 21
For n = 5: a5 = 4(5) + 5 = 25
Therefore, the first five terms of the sequence are: 9, 13, 17, 21, 25.
To find the sum of the first 25 terms, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(2a + (n-1)d)
Here, n is the number of terms, a is the first term, and d is the common difference. In this case, the first term a1 = 9 and the common difference d = 4.
Substituting these values into the formula:
S25 = (25/2)(2(9) + (25-1)(4))
= (25/2)(18 + 24)
= (25/2)(42)
= 525
Therefore, the sum of the first 25 terms of the sequence is 525.
To determine if the sequence an = 4n + 5 is an arithmetic sequence, we need to check if there is a common difference between consecutive terms. In this case, the common difference is 4, which means the difference between any two consecutive terms is always 4.
Thus, the sequence an = 4n + 5 is an arithmetic sequence since it has a constant common difference of 4.