Question

The first term of an A.P. is 6 and the common difference is 5. Find the A.P. and its general term.

Answer 1

Hint- In this question, we just have to find out the successive terms using the first term and the common difference and thus we can also write the general term on the basis of the pattern so found.

Complete step-by-step solution -
Here first term, a=6 and the common difference, d=5.
Now let’s find out all the other terms.
Since in an A.P. every two consecutive terms have the same common difference.
So, ${a_2} - a = d$, where ${a_2}$ is the second term.
On substituting the value of a and d as 6 and 5 respectively in the equation we have, ${a_2} - 6 = 5$.
So, now ${a_2} = 11$ and in a generalized way we can say that ${a_2} = a + d$.
Follow the same pattern for the ${a_3}$ such that ${a_3} - {a_2} = d$ and on substituting the generalized value of \[{a_2}\] here we get ${a_3} = a + 2d$ and on substituting the values of a and d given, ${a_3} = 6 + 2 \times 5$,
${a_3} = 16$
Again, follow the same pattern for the ${a_4}$ such that ${a_4} - {a_3} = d$ and on substituting the generalized value of \[{a_3}\] here we get ${a_4} = a + 3d$ and on substituting the values of a and d given, ${a_4} = 6 + 3 \times 5$,
${a_4} = 21$
Thus, the given A.P. is 6, 11, 16, 21, ………….
Now on considering the pattern of the terms in a generalized way like for example, for the third term the formula is ${a_3} = a + 2d$and for the fourth term the formula is ${a_4} = a + 3d$. We get to know that for the nth term the formula will be equal to ${a_n} = a + \left( {n - 1} \right)d$ which is our required general term.

Note- For such questions just find the patterns according to the consecutive differences among the terms and use the pattern to obtain the required A.P. and also generalize the nth term on the basis of that and also we can check whether the required A.P. is right or not by substituting the values given in the nth term equation.