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

Hint: 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.