How do you write the first six terms in the following sequence \[{a_n} = 3n + 1\]?

Question

How do you write the first six terms in the following sequence $${a_n} = 3n + 1$$?

Vedantu Content Team · Accepted Answer

Hint: The sequence of each term gives us many different values. The sequence $${a_n}$$ has a n number of values, when we put different values to the sequence it gives a different number.Now we are going to substitute positive integers for n of order $$n = 1,2,3,4,5,6$$. Complete step-by-step solution:Now given sequence is $${a_n} = 3n + 1$$,Take $$n$$ values for 6 as we are asked to find the first 6 terms, so, $$n = 1,2,3,4,5,6$$,Now substituting each values we get,Now we take $$n = 1$$,$${a_n} = 3n + 1$$,Substituting $$n = 1$$ in the sequence we get,$$ \Rightarrow {a_1} = 3\left( 1 \right) + 1$$,Now simplifying we get,$$ \Rightarrow {a_1} = 3 + 1$$,Now adding we get,$$ \Rightarrow {a_1} = 4$$So the first term is 4.Now we take$$n = 2$$,$${a_n} = 3n + 1$$,Substituting $$n = 2$$ in the sequence we get,$$ \Rightarrow {a_{}} = 3\left( 2 \right) + 1$$,Now simplifying we get,$$ \Rightarrow {a_2} = 6 + 1$$,Now adding we get,$$ \Rightarrow {a_2} = 7$$,So the second term is 7.Now we take $$n = 3$$,$${a_n} = 3n + 1$$,Substituting $$n = 3$$ in the sequence we get,$$ \Rightarrow {a_3} = 3\left( 3 \right) + 1$$,Now simplifying we get,$$ \Rightarrow {a_3} = 9 + 1$$,Now adding we get,$$ \Rightarrow {a_3} = 10$$,So the third term is 10.Now we take $$n = 4$$,$${a_n} = 3n + 1$$,Substituting $$n = 4$$ in the sequence we get,$$ \Rightarrow {a_4} = 3\left( 4 \right) + 1$$,Now simplifying we get,$$ \Rightarrow {a_4} = 12 + 1$$,Now adding we get,$$ \Rightarrow {a_4} = 13$$,So the third term is 13.Now we take $$n = 5$$,$${a_n} = 3n + 1$$,Substituting $$n = 5$$ in the sequence we get,$$ \Rightarrow {a_5} = 3\left( 5 \right) + 1$$,Now simplifying we get,$$ \Rightarrow {a_5} = 15 + 1$$,Now adding we get,$$ \Rightarrow {a_5} = 16$$,So the fifth term is 16.Now we take $$n = 6$$,$${a_n} = 3n + 1$$,Substituting $$n = 6$$ in the sequence we get,$$ \Rightarrow {a_6} = 3\left( 6 \right) + 1$$,Now simplifying we get,$$ \Rightarrow {a_6} = 18 + 1$$,Now adding we get,$$ \Rightarrow {a_6} = 19$$,So the sixth term is 19.The first 6 terms for the sequence are 4, 7, 10, 13, 16, and 19.$$\therefore $$The six terms of the sequence $${a_n} = 3n + 1$$ are 4, 7, 10, 13, 16, and 19.Note: A sequence is an ordered list of numbers.Example: 1, 3, 5, 7, 9…….The three dots mean to continue forward in the pattern established. Each number in the sequence is called a term. In the sequence 1, 3, 5, 7, 9, …, 1 is the first term, 3 is the second term, 5 is the third term, and so on. The notation $${a_1},{a_2},{a_3}........{a_n}$$ is used to denote the different terms in a sequence.The expression $${a_n}$$ is referred to as the general or $n^{th}$ term of the sequence.