If the $\displaystyle p^{th}$ term of an arithmetic progression is $\displaystyle \frac{1}{q}$and the $\displaystyle 9^{th}$ term $\displaystyle = \frac{1}{p}$,show that $\displaystyle (pq)^{th}$ is $\displaystyle 1$
Books are not written by the great Authority. They are written by humans.
Prove it out. An arithmetic progression has a starting value (call it 'a') and a constant difference (call it 'd').
"the pth term is 1/q"
a + (p-1)d = 1/q
"the 9th term is 1/p"
a + (9-1)d = 1/p
This leads to $\displaystyle p\cdot q = \frac{1}{(a+8d)(a-d)+d}$
p*q is going to have to be an integer. You tell me how this can be and then the implications of the result.
Hello, mastermin346!
I believe there is a typo . . . It is not the 9th term.
. . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $\displaystyle \downarrow$
$\displaystyle \text{If the }p^{th}\text{ term of an arithmetic progression is }\tfrac{1}{q}\,\text{ and the }q^{th}\text{ term is }\tfrac{1}{p}$
$\displaystyle \text{show that the }(pq)^{th}\text{ term is }1.$
$\displaystyle \text{Note that: }\,p \ne q$
$\displaystyle \text{The }n^{th}\text{ term is: }\:a_n \:=\:a + (n-1)d$
. . $\displaystyle \text{where }a\text{ = first term, }d\text{ = common difference.}$
$\displaystyle \text{We are told: }\:\begin{Bmatrix} a_p \:=\:\frac{1}{q} & \Rightarrow & a + (p-1)d &=& \frac{1}{q} & [1] \\ \\[-3mm] a_q \:=\:\frac{1}{p} & \Rightarrow & a + (q-1)d &=& \frac{1}{p} & [2] \end{Bmatrix}$
. . $\displaystyle \begin{array}{ccccc}\text{From [1]:} & aq + dpq - dq \:=\:1 & [3] \\ \\[-3mm] \text{From [2]:} & ap + dpq - dp \:=\:1 & [4] \end{array}$
$\displaystyle \text{Equate [3] and [4]: }\:aq + dpq - dq \:=\:ap + dpq - dp $
. . $\displaystyle ap - aq - dp + dq \:=\:0 \quad\Rightarrow\quad a(p-q) - d(p-q) \:=\:0 $
. . $\displaystyle (p-q)(a-d)\:=\:0 \quad\Rightarrow\quad \rlap{/////}p = q,\;a = d\;\;[5]$
$\displaystyle \text{Substitute [5] into [3]: }\:dq + dpq - dq \:=\:1 \quad\Rightarrow\quad dpq \:=\:1 \quad\Rightarrow\quad pq \:=\:\tfrac{1}{d}$
$\displaystyle \text{Then: }\:a_{pq} \;=\;a_{\frac{1}{d}} \;=\;d + \left(\tfrac{1}{d}-1\right)d \;=\; d + 1 - d$
. . $\displaystyle \text{Therefore: }\:a_{pq} \;=\;1$
This has to be an assumption in the problem statement because nothing prevents $\displaystyle p$th term to be $\displaystyle 1/p$; this does not imply that $\displaystyle p^2$th term is 1.Note that: $\displaystyle p\ne q$