The sums of $n$ terms of three $A.P.'s$ whose first term is $1$ and common differences are $1, 2, 3$ are ${S_1},\;{S_2},\;{S_3}$ respectively. The true relation is

The sum of $n$ terms of two arithmetic progressions are in the ratio $(3 n+8):(7 n+15) .$ Find the ratio of their $12^{\text {th }}$ terms.

In an $A.P.,$ if $p^{\text {th }}$ term is $\frac{1}{q}$ and $q^{\text {th }}$ term is $\frac{1}{p},$ prove that the sum of first $p q$ terms is $\frac{1}{2}(p q+1),$ where $p \neq q$

Find the $17^{\text {th }}$ and $24^{\text {th }}$ term in the following sequence whose $n^{\text {th }}$ term is $a_{n}=4 n-3$

Shamshad Ali buys a scooter for $Rs$ $22000 .$ He pays $Rs$ $4000$ cash and agrees to pay the balance in annual instalment of $Rs$ $1000$ plus $10 \%$ interest on the unpaid amount. How much will the scooter cost him?

If $\log _{3} 2, \log _{3}\left(2^{x}-5\right), \log _{3}\left(2^{x}-\frac{7}{2}\right)$ are in an arithmetic progression, then the value of $x$ is equal to $.....$