Question

What is the pattern in the sequence \[2,5,10,17,28,41,{\text{ }}58,77,100\]

Answer 1

Hint: Observe the numbers carefully. You will find some common ideas between them. First check with the common difference, if it doesn’t work go for a common ratio. If both are not working then apply all other numerical magic.

Complete step-by-step solution:
The given sequence is \[2,5,10,17,28,41,{\text{ }}58,77,100\]
Observe the pattern given..
There is no common difference maintained.
There is no common ratio maintained.
Let’s check term by term for their relation.
The first term \[t_1{\text{ }} = {\text{ }}2\]
The second term \[t_2{\text{ }} = {\text{ }}5\]
And the third term is \[t_3{\text{ }} = {\text{ }}10\]
And so on…
We don’t have any common difference maintained in the given numbers here…
Observe carefully
What relation we can find out..
Lets us add some number to the term 1 and see
For the first term if we add 3 then we get 2nd term
That is ${t_1}+3={t_2}(2+3)=5$
Similarly for the second term if we add 5 then we get third term
That is ${t_2}+5={t_3}(5+5)=10$
Similarly with the fourth term
If we add something we get the next term
Just observe the number which we are adding here
First we added 3, then 5, then 7 then 11…
What are these numbers called ?
\[2,{\text{ }}3,{\text{ }}5,{\text{ }}7{\text{ }},{\text{ }}11,{\text{ }}13\] and so on.. are called prime numbers.
Therefore we come to a conclusion that the nth term of a sequence is sum of the first n prime numbers
Let us check
Nth term suppose \[n = {\text{ }}4\]
4th term = $t_4$ is the sum of the first 4 prime numbers
What are first 4 prime numbers \[2{\text{ }},{\text{ }}3,{\text{ }}5,{\text{ }}7\]
The fourth term of the sequence should be $2+3+5+7=17$ which is satisfied.
Check with $t_5$, $t_6$ and so on….

Note: It is not mandatory that always common difference or common ratio should be maintained. We need to observe carefully for the hidden relation to find out.
Prime number combination, multiple of some number and adding a fixed number, such patterns also you can observe.