Question

\[6,11,16,.....\] Is an arithmetic sequence. What is its next term? What is the least three digit number that comes as a term of this sequence?

Answer 1

Hint: In this question, we have to find out the required value from the given particulars.
We need to first find out the common difference using the formula, then adding it with the last term we can find out the next term. For the second question we need to put the formula greater equals to one hundred and putting common difference and the first term, we can find out the least three digit number that comes as a term of this sequence.

Formula used: Property of Arithmetic progression:
The nth term of the arithmetic sequence is
\[{a_n} = a + \left( {n - 1} \right)d\]
Where,
a = first term of the sequence
d = common difference
n= number of terms

Complete step-by-step solution:
It is given that, \[6,11,16,.....\] is an arithmetic sequence.
We need to find out its next term.
The common difference = second term – first term
\[ \Rightarrow 11 - 6 = 5\]
The next term of the sequence \[ = 16 + 6\]\[ = 21\]
Now we need to find out the least three digit number that comes as a term of this sequence.
We know that, the nth term of the arithmetic sequence is
\[{a_n} = a + \left( {n - 1} \right)d\]
Where,
a = first term of the sequence
d = common difference
n= number of terms
Thus we need to find out the value of n for which,
\[a + \left( {n - 1} \right)d \geqslant 100\]
Substituting the values in given,
\[ \Rightarrow 6 + \left( {n - 1} \right) \times 5 \geqslant 100\]
Multiplying we get,
\[ \Rightarrow 6 + 5n - 5 \geqslant 100\]
Subtracting the terms we get,
\[ \Rightarrow 5n + 1 \geqslant 100\]
Simplifying we get,
\[ \Rightarrow n \geqslant \dfrac{{99}}{5}\]
Hence,
\[ \Rightarrow n \geqslant 19.8\]
Hence the value of n is \[20\].
Therefore, the least three digit number that comes as a term of this sequence is \[6 + \left( {20 - 1} \right) \times 5\]
\[ \Rightarrow 6 + 19 \times 5\]
\[ \Rightarrow 101\]

$\therefore $ The least three digit number that comes as a term of this sequence is \[101\].

Note: We can observe that from the solution, an arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant. In General we write an Arithmetic Sequence like this: \[\left\{ {a,a + d,a + 2d,a + 3d,.......} \right\}\]
Where $a$ the first term, and d is the difference between the terms (called the "common difference").