Question

If an A.P is given by 7, 12, 17, 22, then the ${{n}^{th}}$ term of the A.P is
[a] $2n+5$
[b] $4n+3$
[c] $5n+2$
[d] $3n+4$

Answer 1

Hint: Use the fact that if a is the first term of an A.P and d is the common difference of the A.P, then the ${{n}^{th}}$ term of the A.P is given by ${{a}_{n}}=a+\left( n-1 \right)d$. Find the first term and the common difference of the A.P and hence find the nth term of the A.P.

Complete step-by-step solution -
Here the A.P is 7, 12 , 17, 22, …
Hence, the first term of the A.P is 7 and the common difference of the A.P is 12-7 = 5
Hence, we have a= 7 and d = 5.
We know that if a is the first term of an A.P and d is the common difference of the A.P, then the ${{n}^{th}}$ term of the A.P is given by ${{a}_{n}}=a+\left( n-1 \right)d$.
Hence, we have
${{a}_{n}}=7+\left( n-1 \right)5$
Using the distributive property of multiplication over subtraction, we get
${{a}_{n}}=7+5n-5=5n+2$
Hence the nth term of the A.P is given by 5n+2
Hence option [c] is correct.

Note: Alternative Solution:
We check option wise for n = 1, n=2 and verify which of the options gives correct results of the first terms and the second term.
Checking option [a]:
Put n = 1, we get
2(1) +5 = 7. Hence option [a] holds true for n =1
Put n = 2, we get
2(2)+5 =5. But the second term of the A.P is 12.
Hence option [a] is incorrect.
Checking option [b]:
Put n = 1, we get
4(1) +3 = 7. Hence option [b] holds true for n =1
Put n = 2, we get
4(2)+3 =11. But the second term of the A.P is 12.
Hence option [b] is incorrect.
Checking option [c]:
Put n = 1, we get
5(1) +2 = 7. Hence option [c] holds true for n =1
Put n = 2, we get
5(2)+2 =12.
Hence option [c] is correct.
Checking option d:
Put n = 1, we get
3(1) +4 = 7. Hence option [d] holds true for n =1
Put n = 2, we get
3(2)+5 =8. But the second term of the A.P is 12.
Hence option [d] is incorrect.