Question

Write the correct number if in the given boxes on the basis of the following A.P.
$- 3, - 8, - 13, - 18,....$
Here ${t_3} =$ $?$ , ${t_2} = ?$ , ${t_4} = ?$ , ${t_1} = ?$ , ${t_2} - {t_1} = ?$ , $t{}_3 - {t_2} = ?$
Therefore $a = ?$ , $d = ?$

Answer 1

First we have to define what the terms we need to solve the problem are.
An arithmetic progression can be given by $a,(a + d),(a + 2d),(a + 3d),...$ where $a$ is the first term and $d$ is the common difference.
$a,b,c$ are said to be in arithmetic progression if the common difference between any two-consecutive number of the series is same that is $b - a = c - b \Rightarrow 2b = a + c$

Complete step by step answer:
Formula to consider for solving these questions ${a_n} = a + (n - 1)d$
Where $d$ is the common difference, $a$ is the first term, since we know that difference between consecutive terms is constant in any A.P
Here the given question is $- 3, - 8, - 13, - 18,....$
As we clearly $a = - 3$ is the first term in the given arithmetic progression also ${t_1} = - 3$ is the starting value too, and then the second term is $- 8$ and thus ${t_2} = - 8$
So, since we know the values for ${t_1},{t_2}$ we further proceed to find ${t_2} - {t_1} = ?$ which is ${t_2} - {t_1} = - 8 - ( - 3)$
And thus ${t_2} - {t_1} = -5$ and now the third term is $- 13$ and thus $t{}_3 - {t_2} = - 13 - ( - 8)$ $= - 5$ which is the common difference of the values,
Hence as we see the common difference is $- 5$ .also the fourth term is $- 18$ from the given problem.
Therefore, we get all the values which is ${t_3} = -13$ , ${t_2} = - 8$ , ${t_4} = - 18$ , ${t_1} = - 3$
Also ${t_2} - {t_1} = - 5$ , $t{}_3 - {t_2} = - 5$ and $a = - 3$ , $d = - 5$ , thus we find all unknown values using the arithmetic progression.

Note: To solve most of the problems related to AP, the terms can be conveniently taken as
$3$ Terms; $(a - d),a,(a + d)$
$4$ Terms; $(a - 3d),(a - d),(a + d),(a + 3d)$
If each term of an AP is added, subtracted, multiplied or divided by the same non-zero constant,
The resulting sequence also will be in AP. In an arithmetic progression, the sum of terms from beginning and end will be constant.