Question

In an A.P, if the common difference is 4 and the seventh term is 4, then find the first term.

Answer 1

Hint: First of all write the 7th term of A.P using the formula of the nth term that is \[{{a}_{n}}=a+\left( n-1 \right)d\]. Then equate it to 4 and then substitute the value of ‘d’ in it to find the value of ‘a’ which is the first term of A.P.



Complete step-by-step answer:
We are given an A.P which has a common difference 4 and its seventh term is also 4. We have to find the first term of this A.P.
Let us first understand what Arithmetic Progression (A.P) is. Arithmetic Progression is the series of numbers so that the difference of any two successive numbers is a constant value. For example, the series of even numbers: 2, 4, 6, 8 are in A.P with the first term as 2 and common difference as 2. Also, the nth term of A.P is \[{{a}_{n}}=a+\left( n-1 \right)d\] where ‘a’ is the first term, and ‘d’ is the common difference.
Now, we know that the nth term of any arithmetic progression is, \[{{a}_{n}}=a+\left( n-1 \right)d\] where ‘a’ is the first term and ‘d’ is the common difference.
By substituting n = 7 in the above equation, we get,
\[{{a}_{7}}=a+\left( 7-1 \right)d\]
Or, \[{{a}_{7}}=a+6d\]
We are given that the seventh term of A.P is 4. So, by substituting \[{{a}_{7}}=4\] in the above equation, we get,
\[4=a+6d\]
Also, we are given that the common difference of this A.P is 4. So, by substituting d = 4 in the above equation, we get,
\[4=a+6\left( 4 \right)\]
Or, \[24+a=4\]
By subtracting 24 from both sides of the above equation, we get,
\[24+a-24=4-24\]
So, we get a = – 20
Hence, we get the first term of A.P as – 20.

Note: Some students make this mistake of writing the 7^th term as (a + 7d) directly without considering the formula for the nth term. But this is wrong. They must consider the formula for an nth term first that is \[{{a}_{n}}=a+\left( n-1 \right)d\] and then substitute in it the value of n = 7 to get the 7^th term that is \[{{a}_{7}}=a+6d\]. So, proper care must be taken while writing any term.