Q.If $\dfrac{a^p+b^p}{a^{p-1}+b^{p-1}}$ is the A.M. between $a$ and $b$, then find the value of $p$. Also, for $p, q, r$ in A.P., prove that $(a-b)r + (b-c)p + (c-a)q = 0$. [Note: the book's expression appeared truncated at the page break; the identity is completed with "$= 0$" as strongly implied by the "prove that" phrasing and the standard result.]

Introduction — What It Means and Why It Matters

You've been solving problems your whole life. When you see a new type of question, you first figure out what it's asking, then decide which tools to use, then carry out the steps. That first step — figuring out what's going on — is the heart of an introduction in mathematics and science.

An introduction is not a summary. It's a setup. It tells you the players, the rules, and the goal before the action starts.

The Intuition

Imagine you walk into a room and see a chessboard with pieces already arranged. Without knowing whose turn it is, what the last move was, or whether you're playing standard chess or a variant, you can't make sense of anything. An introduction is like being told: "White to move, standard rules, checkmate in two."

In a proof, an introduction declares the assumptions you're allowed to use. In a problem, it states the given data and the unknown you're after. In a definition, it names the new object and lists its essential properties.

The Precise Statement

In formal logic and mathematics, an introduction rule tells you how to legitimately bring a new statement into a derivation. The most common example is the conditional introduction (also called "direct proof" or "→-introduction"):

If from assumption $P$ you can derive $Q$, then you may conclude $P \implies Q$.

That is, to prove "if $P$ then $Q$", you temporarily assume $P$ is true, work through the reasoning, and arrive at $Q$. Then you discharge the assumption and write $P \implies Q$ as a proven statement.

Symbolically:

The line means "from the top, infer the bottom." The dots represent a valid chain of reasoning.

Why This Matters for Exams

When you see a question that asks you to "prove that if $x > 2$ then $x^2 > 4$", you are being asked to perform a conditional introduction. You start by introducing the assumption $x > 2$, work from there, and end with $x^2 > 4$. Then you write the implication. …