If the ratio of the sum of the first $m$ terms of an A.P. to the sum of the first $n$ terms is $m^2:n^2$, find the ratio of its 19th term to its 21st term.

> [!TLDR] The given ratio of sums forces $d=2a$ in the A.P., which yields $a_{19}:a_{21}=37:41$. > [!FORMULA] > Sum of $n$ terms: $S_n=\dfrac n2\big[2a+(n-1)d\big]$. $n$th term: $a_n=a+(n-1)d$. 1. $\dfrac{S_m}{S_n}=\dfrac{\frac m2[2a+(m-1)d]}{\frac n2[2a+(n-1)d]}=\dfrac{m[2a+(m-1)d]}{n[2a+(n-1)d]}=\dfrac{m^2}{n^2}$ (given). 2. This simplifies to $\dfrac{2a+(m-1)d}{2a+(n-1)d}=\dfrac mn$ for every choice of $m,n$. 3. Define $f(k)=2a+(k-1)d$. The relation $\dfrac{f(m)}{f(n)}=\dfrac mn$ for all $m,n$ means $f(k)=ck$ for some constant $c$. 4. At $k=1$: $f(1)=2a=c(1)\Rightarrow c=2a$. So $2a+(k-1)d=2ak$ for every $k$. 5. Taking $k=2$: $2a+d=4a\Rightarrow d=2a$. …

If the ratio of the sum of the first m terms of an A.P. to the sum of the first n terms is m²:n², find the rat

Exercise 5.1 · Q19

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Concept understanding — Introduction

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.

Note

The word "introduction" here is about logical introduction — bringing a new idea or assumption into a discussion — not the first paragraph of an essay.

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:

\frac{ \begin{array}{c} P \\ \vdots \\ Q \end{array} }{ P \implies Q }

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. …

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The given ratio of sums forces $d=2a$ in the A.P., which yields $a_{19}:a_{21}=37:41$ .

n

$\dfrac{S_m}{S_n}=\dfrac{\frac m2[2a+(m-1)d]}{\frac n2[2a+(n-1)d]}=\dfrac{m[2a+(m-1)d]}{n[2a+(n-1)d]}=\dfrac{m^2}{n^2}$ (given).
This simplifies to $\dfrac{2a+(m-1)d}{2a+(n-1)d}=\dfrac mn$ for every choice of $m,n$ .
Define $f(k)=2a+(k-1)d$ . The relation $\dfrac{f(m)}{f(n)}=\dfrac mn$ for all $m,n$ means $f(k)=ck$ for some constant $c$ .
At $k=1$ : $f(1)=2a=c(1)\Rightarrow c=2a$ . So $2a+(k-1)d=2ak$ for every $k$ .
Taking $k=2$ : $2a+d=4a\Rightarrow d=2a$ . …

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Q.If the ratio of the sum of the first mmm terms of an A.P. to the sum of the first nnn terms is m2:n2m^2:n^2m2:n2, find the ratio of its 19th term to its 21st term.

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Q.If the ratio of the sum of the first $m$ terms of an A.P. to the sum of the first $n$ terms is $m^2:n^2$ , find the ratio of its 19th term to its 21st term.

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