Answer:
The smallest integer is 9 and there are 19 terms in the sequence.
Step-by-step explanation:
Arithmetic Sequence
The general term of an arithmetic sequence is
[tex]\displaystyle a_n=a_1+(n-1)r\ ........[eq\ 1][/tex]
And the sum of all n terms is
[tex]\displaystyle s_n=\frac{a_1+a_n}{2}n...... [eq\ 2][/tex]
The sequence of the question complies with
[tex]\displaystyle s_n=342[/tex]
[tex]\displaystyle a_n=3a_1[/tex]
Using the last condition in eq 1 and knowing that r=1 (consecutive numbers)
[tex]\displaystyle a_n=a_1+n-1=3a_1[/tex]
Rearranging
[tex]\displaystyle 2a_1=n-1[/tex]
Using eq 2
[tex]\displaystyle \frac{a_1+a_n}{2}n=342[/tex]
Replacing the first condition
[tex]\displaystyle \frac{a_1+3a_1}{2}n=342[/tex]
Simplifying
[tex]\displaystyle 2a_1\ n=342[/tex]
Since
[tex]\displaystyle 2a_1=n-1[/tex]
We have
[tex]\displaystyle n(n-1)=342[/tex]
Factoring
[tex]\displaystyle n(n-1)=(19)(18)[/tex]
We find the number of terms
[tex]\displaystyle n=19[/tex]
The first term is
[tex]\displaystyle a_1=\ \frac{342}{38}=9[/tex]
