Question

How do you find the next three terms of the arithmetic sequence \[ - 2, - 5, - 8, - 11............\]?

Answer 1

Hint: There are mainly two types of sequences
Arithmetic sequences: This is a type of sequence where the difference between consecutive terms is always the same.
Geometric sequence: This is a type of sequence where the ratio between consecutive terms is always the same.
Now with the help of the above two definitions we will find the given series and accordingly let us solve the problem.

Complete step by step solution:
Given
\[ - 2, - 5, - 8, - 11............\;\;\;\;.............\left( i \right)\]
We have been given the above arithmetic sequence and we need to find the next 3 terms of it.
Now we know that:
Arithmetic sequences:
This is a type of sequence where the difference between consecutive terms is always the same and we call that difference a common difference.
$d = {\text{difference of successive terms}}$
Also the general term for an arithmetic sequence is given by:
${a_n} = a + \left( {n - 1} \right)d$
So first let’s find its common difference of the successive terms of the given sequence \[ - 2, - 5, - 8, - 11............\] such that, we can write:
$
\Rightarrow d = - 5 - \left( { - 2} \right) = - 5 + 2 = - 3 \\
\Rightarrow d = - 8 - \left( { - 5} \right) = - 8 + 5 = - 3 \\
\Rightarrow d = - 11 - \left( { - 8} \right) = - 11 + 8 = - 3 \\
 $
So on observing the above results we can conclude that the common difference:
$d = - 3$
Now we have been given \[{a_1},{a_2},{a_3}\;{\text{and}}\;{a_4}\], and we need to find${a_5},{a_6}\;and\;{a_7}$.
For that we have the equation ${a_n} = a + \left( {n - 1} \right)d$:
Such that we can write:
\[
\Rightarrow {a_5} = a + \left( {n - 1} \right)d \\
 \Rightarrow - 2 + \left( {5 - 1} \right) \times - 3 \\
   \Rightarrow - 2 + 4 \times - 3 \\
   \Rightarrow - 2 - 12 \\
   \Rightarrow - 14.....................................\left( {ii} \right) \\
  \Rightarrow {a_6} = a + \left( {n - 1} \right)d \\
  \Rightarrow - 2 + \left( {6 - 1} \right) \times - 3 \\
   \Rightarrow - 2 + 5 \times - 3 \\
   \Rightarrow - 2 - 15 \\
   \Rightarrow - 17.....................................\left( {iii} \right) \\
\Rightarrow {a_7} = a + \left( {n - 1} \right)d \\
   \Rightarrow - 2 + \left( {7 - 1} \right) \times - 3 \\
   \Rightarrow - 2 + 6 \times - 3 \\
   \Rightarrow - 2 - 18 \\
   \Rightarrow - 20.....................................\left( {iv} \right) \\
 \]

Therefore from (ii), (iii) and (iv) we can write the next three terms of the arithmetic sequence \[ - 2, - 5, - 8, - 11............\] \[{a_5},{a_6}\;,{a_7}\] as ${a_5} = - 14,{a_6} = - 17,{a_7} = - 20$.

Note: General terms of arithmetic and geometric series:
Arithmetic series: ${a_n} = a + \left( {n - 1} \right)d$
Geometric series:${a_n} = r \times {a_{n - 1}}$
While doing similar questions one should be very thorough with the properties and formulas regarding the sequences and care must be given to minimize the errors in the calculation process.