Answer:
\boxed{a_{8} = 36}
a
8
=36
Step-by-step explanation:
By determining the pattern, we will be able to compute for the missing terms in the sequence.
In this case we can see that the increase is +2, +3, +4, +5.
1+2=3
3+3=6
6+4=10
10+5=15
Let:
a_{1} = 1a
1
=1
a_{2} = 3a
2
=3
a_{3} = 6a
3
=6
a_{4} = 10a
4
=10
a_{5} = 15a
5
=15
FORMULA FOR THE SEQUENCE PATTERN:
a_{n} = a_{n-1}+na
n
=a
n−1
+n
To check if it really works, the following tests have been made:
n=2
\begin{gathered} a_{2} = a_{2-1}+2 \\ = a_{1}+2 \\ = 1+2 \\ a_{2} = 3 \end{gathered}
a
2
=a
2−1
+2
=a
1
+2
=1+2
a
2
=3
n=3
\begin{gathered} a_{3} = a_{3-1}+3 \\ = a_{2}+3 \\ = 3+3 \\ a_{3} = 6 \end{gathered}
a
3
=a
3−1
+3
=a
2
+3
=3+3
a
3
=6
n=4
\begin{gathered} a_{4} = a_{4-1}+4 \\ = a_{3}+4 \\ = 6+4 \\ a_{4} = 10 \end{gathered}
a
4
=a
4−1
+4
=a
3
+4
=6+4
a
4
=10
n=5
\begin{gathered} a_{5} = a_{5-1}+5 \\ = a_{4}+5 \\ = 10+5 \\ a_{5} = 15 \end{gathered}
a
5
=a
5−1
+5
=a
4
+5
=10+5
a
5
=15
Having done those, we will now compute for the missing terms in the sequence:
n=6
\begin{gathered} a_{6} = a_{6-1}+6 \\ = a_{5}+6 \\ = 15+6 \\ a_{6} = 21 \end{gathered}
a
6
=a
6−1
+6
=a
5
+6
=15+6
a
6
=21
n=7
\begin{gathered} a_{7} = a_{7-1}+7 \\ = a_{6}+7 \\ = 21+7 \\ a_{7} = 28 \end{gathered}
a
7
=a
7−1
+7
=a
6
+7
=21+7
a
7
=28
n=8
\begin{gathered} a_{8} = a_{8-1}+8 \\ = a_{7}+8 \\ = 28+8 \\ \boxed{a_{8} = 36} \end{gathered}
a
8
=a
8−1
+8
=a
7
+8
=28+8
a
8
=36
It's also possible to get the value with
\boxed{ a_{n} = \frac{(1+n)×n}{2} }
a
n
=
2
(1+n)×n
n=8
a_{8} = \frac{(1+8)×8}{2} = \frac{9×8}{2} = \frac{72}{2} = 36a
8
=
2
(1+8)×8
=
2
9×8
=
2
72
=36
a_{8} = 36a
8
=36
