Question

How do you complete the statement for the arithmetic sequence: 170 is the _ term of $-4,2,8,...$?

Answer 1

Hint: We are given with an arithmetic sequence using we will find the properties of the sequence. Firstly, we will assign \[{{a}_{1}}=-4\], \[{{a}_{2}}=2\] and \[{{a}_{3}}=8\]. Then we will find the difference between the consecutive term by subtracting second term by first term or by subtracting third term by second term and we get it as 6. Then, we substitute the values known in the nth term formula, that is, \[{{a}_{n}}=a+(n-1)d\]. On solving, we will get the value of ‘n’ for which \[{{a}_{n}}=170\].

Complete step by step solution:
According to the given question, we are given an arithmetic sequence, using which we have to calculate the value of ‘n’ for which the \[{{a}_{n}}=170\].
Arithmetic sequence can be said to be a set of numbers having the same difference between each consecutive numbers.
\[{{a}_{n}}=a+(n-1)d\]
Here, \[{{a}_{n}}\]refers to the nth term
\[a\] refers to the first term of the sequence
\[n\] refers to the position of a term in the sequence
\[d\] is the common difference between each consecutive term.

Let us assign the given terms as,
\[{{a}_{1}}=a=-4\]
\[{{a}_{2}}=2\]
\[{{a}_{3}}=8\]
Now, we have to find the common difference between each consecutive term, \[d\].
We can find this by subtracting second term by first term or third term by second term, we have,
\[d={{a}_{2}}-{{a}_{1}}\]
\[\Rightarrow d=2-(-4)\]
\[\Rightarrow d=2+4\]
\[\Rightarrow d=6\]
We know that, the formula of nth term of an arithmetic sequence is \[{{a}_{n}}=a+(n-1)d\].
Now, we will substitute the known values in the above formula and we get,
\[{{a}_{n}}=a+(n-1)d\]
\[\Rightarrow 170=(-4)+(n-1)6\]
Solving for ‘n’ we get,
Adding 4 on both sides, we have,
\[\Rightarrow 170+4=(-4)+(n-1)6+4\]
Proceeding with the calculations we have,
\[\Rightarrow 174=(n-1)6\]
Now, multiplying by \[\dfrac{1}{6}\] on both the sides, we have,
\[\Rightarrow 174\times \dfrac{1}{6}=(n-1)6\times \dfrac{1}{6}\]
\[\Rightarrow 29=n-1\]
\[\Rightarrow n=30\]
Therefore, 170 is the 30th term of the given arithmetic sequence.

Note: The arithmetic sequence has a constant difference between the consecutive terms, so if the question is asked to check if the sequence is an arithmetic or not, simply check the difference between each of the given consecutive terms. The nth term formula is only valid for arithmetic sequence, so before using the formula check whether the sequence is an arithmetic sequence or not.