Question

Solve the following problems related to sequences. Show all of your work and clearly indicate your final answer. a) Find the 7th term in the sequence: -19,-9, 1, ... b) Find the 8th term in the sequence: -3, -9,-27, ... c) Find the common difference and use it to fill in the missing terms in this arithmetic sequence. ..., -32, , -12,

Answer 1

The sequence of numbers is an arithmetic sequence. The sequence,

`-19,-9,1`

has a common difference of 10.

This is derived by deducting each term from the next one. That is deduct the first term from the second term, deduct the second term from the third, and so on. So you would have,

`\begin{gathered} d=-9-\lbrack-19\rbrack \\ d=-9+19 \\ d=10 \end{gathered}`

To find the nth term, we shall apply the formula;

`\begin{gathered} n_(th)=a+(n-1)d \\ a=-19,d=10 \\ To\text{ find the 7th term (n=7)} \\ 7_(th)=-19+(7-1)10 \\ 7_(th)=-19+(6)10 \\ 7_(th)=-19+60 \\ 7_(th)=41 \end{gathered}`

The 7th term in the sequence is 41

For the sequence,

-3, -9, -27...

What we have is a geometric sequence and the common ratio is derived by dividing each term by the next one. So you would have

`\begin{gathered} r=(-9)/(-3) \\ r=3 \\ \text{Also,} \\ r=(-27)/(-9) \\ r=3 \end{gathered}`

To find the nth term of a geometric sequence;

`\begin{gathered} N_(th)=ar^(n-1) \\ a=-3,r=3 \\ To\text{ find the 8th term (N=8)} \\ 8_(th)=-3*(3^(8-1)) \\ 8_(th)=-3*(3^7) \\ 8_(th)=-3*2187 \\ 8_(th)=-6561 \end{gathered}`