In an $\mathrm{A.P.}$ if $m^{\text {th }}$ term is $n$ and the $n^{\text {th }}$ term is $m,$ where $m \neq n$, find the ${p^{th}}$ term.
We have $a_{m}=a+(m-1) d=n,$ ......$(1)$
and $\quad a_{n}=a+(n-1) d=m$ .........$(2)$
Solving $(1)$ and $(2),$ we get
$(m-n) d=n-m,$ or $d=-1,$ ...........$(3)$
and $\quad a=n+m-1$ ...........$(4)$
Therefore $\quad a_{p}=a+(p-1) d$
$=n+m-1+(p-1)(-1)=n+m-p$
Hence, the $p^{\text {th }}$ term is $n+m-p$
Suppose $a_{1}, a_{2}, \ldots, a_{ n }, \ldots$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms of the sum of first nine terms of the progression is $5: 17$ and $110< a_{15} < 120$ , then the sum of the first ten terms of the progression is equal to -
If $\log _e \mathrm{a}, \log _e \mathrm{~b}, \log _e \mathrm{c}$ are in an $A.P.$ and $\log _e \mathrm{a}-$ $\log _e 2 b, \log _e 2 b-\log _e 3 c, \log _e 3 c-\log _e a$ are also in an $A.P,$ then $a: b: c$ is equal to
Let $a_1, a_2, a_3, \ldots$ be an arithmetic progression with $a_1=7$ and common difference $8$ . Let $T_1, T_2, T_3, \ldots$ be such that $T_1=3$ and $T_{n+1}-T_n=a_n$ for $n \geq 1$. Then, which of the following is/are $TRUE$ ?
$(A)$ $T_{20}=1604$
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