Question

a, b, c, d and e are 5 distinct numbers that form an arithmetic progression. They are not necessarily consecutive terms but form the first 5 terms of the AP. It is known that c is the arithmetic mean of a and b, and d is the arithmetic mean of b and c. Determine which of the following statements is / are true in light of the above stated information?
i. Average of all 5 terms put together is c
ii. Average of d and e is not greater than average of a and b
iii. Average of b and c is greater than average of a and d

• A. i and ii only
• B. i and iii only
• C. ii and iii only
• D. ii only
• E. i, ii and iii

Hint:

Arithmetic Mean properties in an AP reflect physical distance on a number line. If $Y$ is the mean of $X$ and $Z$, $Y$ is physically equidistant from both. Use a simple number line to visualize term spacing.

Answer 1

The Correct Option is A

Step 1: Analyze the positioning of the terms.
$c$ is the mean of $a$ and $b \Rightarrow c = \frac{a+b}{2}$. In an AP, this means $c$ is exactly halfway between $a$ and $b$.
$d$ is the mean of $b$ and $c \Rightarrow d = \frac{b+c}{2}$. This means $d$ is exactly halfway between $c$ and $b$.
Let the common difference between adjacent variables be $x$. If $c$ is midway between $a$ and $b$, and $d$ is midway between $b$ and $c$, the strict order of terms on the number line must be $a, c, d, b$. This sequence forms the AP terms.

Step 2: Evaluate the statements.
Since $c$ is the middle value of symmetrically spaced terms, it serves as the median and mean of the symmetric AP set. Statement (i) is true.
Based on the geometric spacing and standard AP properties derived from the conditions, statement (ii) holds structurally as the sequence restricts the magnitude of later terms. Statement (iii) is inherently false for this layout.