# Find an for the sequence 1,4,7,10,...

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I believe the term"an" in the question denotes nth term of the given arithmetical sequence.

We can re write the various terms of the given sequence as follows.

First term = a1 = 1 = = 3 - 2 = 3*1 - 2

Second term = a2 = 4 = 6 - 2 = 3*2 - 2

Third term = a3 = 7 = 9 - 2 = 3*3 - 2

Fourth term = a4 = 10 = 12 - 2 = 3*4 - 2

Based on the pattern as emerging from the first four terms of the sequence the nth term can be expressed as:

nth tern = an = 3n - 2

Answer:

an = 3n - 2

Given sequence is 1, 4, 7,10.

a1 = 1, a2 = 4, a3 = 7 , a4 = 10.

We see that a2 -a1 = 4-1 =3. , a3-a2 = 7-4 = 3, a4 - a3 = 10-7 = 3.

Therefore the successive terms have the same difference or common difference d = 3.

The starting term is a1 = 1.

Therefore the n th term , an = a1 + (n-1)d = 1+(n-1)3.

So an = 1+(n-1)3 = 1+3n-3 = 3n-2.

Therefore the nth term an = 3n-2.

We notice that each term is obtained by adding 3 to the preceding term. Therefore, we conclude that the given sequence is an arithmetic sequence, whose common difference is 3.

We'll note the common difference as d = 3.

an is the n-th term of the A.P. and it could be calculated using the formula of general term:

an = a1 + (n-1)*d, where a1 is the first term, n is the number of terms and d is the common difference.

a1 = 1

d = 3

an = 1 + (n-1)*3

We'll remove the brackets and we'll get:

an = 1 + 3n - 3

We'll combine like terms:

an = 3n - 2

Now, we can compute any term of the given sequence:

a1 =1

a2 = 3*2 - 2

a2 = 6-2

a2 = 4

a3 = 3*3 - 2

a3 = 9 - 2

a3 = 7

.........

The formula verifies the terms from the given sequence; 1,4,7,10,...