Question

How do you write a rule for the nth term of the arithmetic sequence and then find $ {a_{10}} $ for $ - 4,2,8,14,20 $ ?

Answer 1

Hint: Sequence is basically a set of things that are in any order. Arithmetic sequence is a sequence where the difference between each successive pair of terms is the same and it is abbreviated as AP. Next term of any sequence can be obtained by adding a constant number to the term before it. That constant number which is added is known as a common difference. Since, all the arithmetic sequences follow the same pattern, we can write the same rule for finding the nth term for the sequence.

Complete step by step solution:
To write the rule for the nth term we need to know the general terms,
  \[
  a = first\;term \\
  d = common\;difference \\
 \]
In general, given the first term and common difference, we can write the following series of equations,
  \[
  {a_2} = {a_1} + d \\
  {a_3} = {a_2} + d = ({a_1} + d) + d = {a_1} + 2d \\
  {a_4} = {a_3} + d = ({a_1} + d) + d = {a_1} + 3d \;
 \]
From the set of above equations, we can derive the final formula as
  \[{a_n} = {a_1} + (n - 1)d\]
We are given the sequence as $ - 4,2,8,14,20 $
  \[
  a = - 4 \\
  d = 6 \;
 \]
  \[{a_n} = -4 + (n - 1)6 \]
We can find the term by putting it in the formula
  \[ \Rightarrow {a_{10}} = - 4 + (10 - 1)6\]
  \[ \Rightarrow {a_{10}} = - 4 + (9)6\]
  \[ \Rightarrow {a_{10}} = - 4 + 54\]
  \[ \Rightarrow {a_{10}} = 50\]
Hence, this is the required answer.

Note: Common difference of any sequence can be obtained by subtracting any of the two terms i.e. subtracting the latter term from prior.
 $ d = {a_n} - {a_{n - 1}} $
Also, Arithmetic Sequence can be both finite and infinite. The behavior of AP depends on the nature of common difference.
I.If the common difference is a positive number, the sequence will progress towards infinity.
II.If the common difference is a negative number, the sequence will regress towards negative infinity.