Question

In which of the arithmetic sequences given below, 37 is a term ?

• A. 4, 9, 14, ....
• B. 7, 12, 17, ....
• C. 8, 12, 16, ....
• D. 7, 11, 15, ....

Hint:

To quickly check if a number is a term in an AP, subtract the first term from the number (aₙ - a) and see if the result is divisible by the common difference d. In option (B), 37 - 7 = 30, and 30 is divisible by the common difference 5. This is a fast way to verify the answer.

Answer 1

The Correct Option is B

We need to determine which of the given arithmetic sequences (APs) contains the number 37 as one of its terms.

The formula for the n-th term of an arithmetic sequence is aₙ = a + (n-1)d, where a is the first term, d is the common difference, and n is the term number.
For 37 to be a term in a sequence, the value of n calculated from the formula must be a positive integer.

Let's check each option:
(A) 4, 9, 14, ....
Here, a = 4 and d = 9 - 4 = 5.
Let aₙ = 37.
37 = 4 + (n-1)5 37 - 4 = (n-1)5 33 = 5(n-1) n-1 = (33)/(5) Since (33)/(5) is not an integer, n will not be an integer. So, 37 is not a term in this sequence.

(B) 7, 12, 17, ....
Here, a = 7 and d = 12 - 7 = 5.
Let aₙ = 37.
37 = 7 + (n-1)5 37 - 7 = (n-1)5 30 = 5(n-1) n-1 = (30)/(5) = 6 n = 6 + 1 = 7 Since n = 7 is a positive integer, 37 is the 7th term of this sequence.

(C) 8, 12, 16, ....
Here, a = 8 and d = 12 - 8 = 4.
Let aₙ = 37.
37 = 8 + (n-1)4 29 = 4(n-1) n-1 = (29)/(4) Since (29)/(4) is not an integer, n will not be an integer. So, 37 is not a term in this sequence.

(D) 7, 11, 15, ....
Here, a = 7 and d = 11 - 7 = 4.
Let aₙ = 37.
37 = 7 + (n-1)4 30 = 4(n-1) n-1 = (30)/(4) = 7.5 Since 7.5 is not an integer, n will not be an integer. So, 37 is not a term in this sequence.

The only sequence for which n is a positive integer is 7, 12, 17, .... Therefore, 37 is a term of this sequence.