Question

Find the common difference (d) of the arithmetic progression. Given that the first term (a) is 3, the total number of terms (n) is 8, and the sum of all the terms (S) is 192.

Answer 1

Hint: According to the question given above, we have to find the common difference (in the arithmetic progression it is denoted by, d. It is always constant and remains the same for the given arithmetic progression. It can be found by the difference between the successive term/number and it’s preceding term/number).
If an A.P. series is $a_1,a_2,a_3,a_4,...................a_n,a_{n-1}$
Common difference can be calculated as $d=(a_2-a_1)=(a_{n-1}-a_2)$
But, for the question given above numbers of terms of an A.P. series are not given. Hence, to find the common difference (d) we have to use the formula to find the sum of all terms which is given in the question S = 192.

Formula used: $S={\displaystyle \frac{n}{2}}\left[2a+\left(n-1\right)d\right]$………………………………..(1)
Where, S is the sum of all the terms, a is the first term, n is the total number of terms and, d is the common difference.

Complete step-by-step answer:
For arithmetic progression given,
First term (a) of A.P. = 3,
Total number of terms (n) of A.P. = 8 and,
Sum of all the terms (S) in the A.P. = 192
Step 1: As given in the question, sum of all terms S = 192 so, with the help of formula (1) we can find the common difference (d).
Step 2: On substituting all the value in the formula ………………..(1)
$192={\displaystyle \frac{8}{2}}\left[2a+\left(8-1\right)d\right]$
Step 3: On solving the equation obtained in step 2.
$$\begin{gathered} 192=4(6+7d) \\ 192=24+28d \\ 192-24=28d \\ 168=28d \end{gathered}$$
After cross-multiplication we can obtain the value of common difference.
$$\begin{gathered} d={\displaystyle \frac{168}{28}} \\ d=6 \end{gathered}$$
Hence, the common difference for the given A.P. is: 6

By using the formula to find the sum of all the terms of A.P. we have obtained the common difference which is $d=6$

Note: Another method: We can also find the common difference (d) with the help of last term (l) (it is the term which comes in the end of the given A.P.) and the sum of all the terms of given A.P.
Step 1: As we know last term of an A.P. is $l=a+(n-1)d$where, $l$is the last term of an A.P.
On substituting all the values in the formula given above,
$$\begin{gathered} l=3+(8-1)d \\ l=3+7d \end{gathered}$$
Step 2: As we know if the last term $l$is given then, sum of all terms of an A.P. can be calculated with the help of formula: $S={\displaystyle \frac{n}{2}}(a+l)$
Step 3: On substituting all the value in the formula mentioned in step 3,
$$\begin{gathered} 192={\displaystyle \frac{8}{2}}(3+6+7d) \\ 192=4(6+7d) \end{gathered}$$
On cross-multiplication we can obtain the value of common difference.
$$\begin{gathered} {\displaystyle \frac{192}{4}}=6+7d \\ 48-6=7d \\ d={\displaystyle \frac{42}{7}} \\ d=6 \end{gathered}$$
Hence, common difference for the given A.P. is $d=6$
An arithmetic progression (A.P.) is defined as a sequence of numbers/digits in which for every pair of consecutive numbers/terms, the second number/term is obtained by a fixed number/term to the first number/term.
A sequence of number is an arithmetic progression if the difference of the term/number and the preceding term is always the same or constant.