Introduction to APs and common difference.

Correct answer: C. No because consecutive differences are not equal

The new terms are (16,81,196), and the differences are (65,115), which are not equal. Squaring generally does not preserve an arithmetic progression.

Correct answer: A. (1)

The original (d=5), and (4n) has (d=4), so the new (d=1). In term-number changes, subtract its difference.

Correct answer: C. Not an arithmetic progression because new differences will not be equal

The new terms are (2,8,16,\ldots), and the differences are (6,8,\ldots), which are not equal. Adding squares of term numbers does not preserve equal difference.

Correct answer: C. \(12\)

Given \(a+3d-a=36\). Cancelling \(a\) gives \(3d=36\), so \(d=36/3=12\). There are three equal gaps, not four, between the first term \(a\) and the fourth term \(a+3d\). Exam tip: In the \(n\)th term \(a+(n-1)d\) of an AP, count the number of gaps between terms carefully.

Correct answer: D. No value is possible

(25-4=21) is split into three equal gaps, so the second term should be (11) and the third (18). This makes (x=11) and (2x+1=18) impossible together.

Correct answer: C. (14)

The selected sequence is (5,19,33,\ldots), and its difference is (14). Selecting every second term makes the new (d) twice the original (d).