# Sum of this sequence: a_1 = 10, a = 3 + a_{n-1}

#### unknownren

##### New member
Explain is it possible to find the sum of the sequence in this problem.

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#### Subhotosh Khan

##### Super Moderator
Staff member
Explain is it possible to find the sum of the sequence in this problem.

Your expression needs to be corrected! You posted:

a1 = 10

$$\displaystyle a_{[?]} \ = \ a_{n-1} + 3$$

On the left-hand-side of the equation there is no refernce to an!

#### unknownren

##### New member
Your expression needs to be corrected! You posted:

a1 = 10

$$\displaystyle a_{[?]} \ = \ a_{n-1} + 3$$

On the left-hand-side of the equation there is no refernce to an!

So its not possible to find the sum of this sequence then? an! is added to the left hand side correct?

#### tkhunny

##### Moderator
Staff member

So its not possible to find the sum of this sequence then? an! is added to the left hand side correct?

To SUM a sequence, it must either END or CONVERGE (in some appropriate sense).

Adding three to each successive term does neither.

#### Dr.Peterson

##### Elite Member
Explain is it possible to find the sum of the sequence in this problem.
Presumably you are asking about finding the sum of the first N terms in the sequence defined by

a1 = 10
an = an-1 + 3

Write out the first several terms of this sequence, and the corresponding sums, in order to get a sense of what you are working with.

Then look up "arithmetic series", and see if you can apply what you learn. Show us whatever you are able to do, and we'll help you out.

#### unknownren

##### New member
Presumably you are asking about finding the sum of the first N terms in the sequence defined by
a1 = 10
an = an-1 + 3

Write out the first several terms of this sequence, and the corresponding sums, in order to get a sense of what you are working with.

Then look up "arithmetic series", and see if you can apply what you learn. Show us whatever you are able to do, and we'll help you out.
a2=3+10=13
a3=3+13=16
a4=3+16=19
a5=3+19=22

a2-a1=13-10=3
a3-a2=16-13=3
a4-a3=19-16=3
a5-a4=22-19=3

common difference is 3

#### Subhotosh Khan

##### Super Moderator
Staff member
a2=3+10=13
a3=3+13=16
a4=3+16=19
a5=3+19=22

a2-a1=13-10=3
a3-a2=16-13=3
a4-a3=19-16=3
a5-a4=22-19=3

common difference is 3
Do you know the sum of first 'n' terms in an arithmatic series with first term 'a' and common difference 'd'?

#### Dr.Peterson

##### Elite Member
a2=3+10=13
a3=3+13=16
a4=3+16=19
a5=3+19=22

a2-a1=13-10=3
a3-a2=16-13=3
a4-a3=19-16=3
a5-a4=22-19=3

common difference is 3
Good; you saw where I was leading you. (Did you notice that the definition itself tells you the common difference: an - an-1 = 3 .)

What more do you know about arithmetic series?

#### pka

##### Elite Member
Explain is it possible to find the sum of the sequence in this problem.
Not being sure of what really what the question is, I take Prof. Peterson's reading.
If $$\displaystyle a_1=10~\&~a_{n}=a_{n-1}+3$$ then $$\displaystyle a_n=3n+7$$ SEE HERE
\displaystyle \begin{align*}S_n&=\sum\limits_{k = 1}^n {{a_k}}\\&=\sum\limits_{k = 1}^n {3k+7}\\&=3\dfrac{n(n+1)}{2}+7n\\&=~? \\ \end{align*}