Question

How do you find the explicit formula for the following arithmetic sequence $123,116,109,102,95$ ?

Answer 1

As seeing the given sequence, clearly it a decreasing series of sequence with a common difference of $-7$ here minus indicates decrement means it is an arithmetic sequence and definitely it will have a linear formula with a variable which depends on the term. Let consider its first term be $a$ and the common difference be $d$ that means the difference between ${{n}^{th}}$ term and first term will be $(n-1)d$ as d is the common difference as all adjacent term has same difference this implies that the ${{n}^{th}}$ term will be the sum of first term and $(n-1)d$.

Complete step-by-step answer:
As in the given sequence, the first term let this be $a$
$\Rightarrow a=123$
And we can see that the difference between all the adjacent term is same, let this be $d$
Since the series is decreasing that means the difference will be conventionally negative
$\Rightarrow d=-7$
Now using the explicit formula for arithmetic sequence
${{a}_{n}}=a+(n-1)d$ , where ${{a}_{n}}={{n}^{th}}\text{ }term\text{ of sequence}$
Substituting all the values in this equation, we will get
$\Rightarrow {{a}_{n}}=123+(n-1)(-7)$
$\Rightarrow {{a}_{n}}=123-(n-1)7$
$\Rightarrow {{a}_{n}}=130-7n$
And clearly the sequence is decreasing
Hence, ${{a}_{n}}=130-7n$.

Note: When we see an arithmetic sequence first find the common difference between the adjacent terms and then just apply the simple common sense that for the ${{n}^{th}}$ term and first term the difference will be $(n-1)d$ and we will get that term by adding the first term in this difference.