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**Hint:**Use the formula of Arithmetic progression sequence for the nth terms that is \[{a_n} = a + \left( {n - 1} \right)d\] where, a initial term of the Arithmetic progression and d is the common difference of successive. Calculate the value of \[{S_1}\].

**Complete step by step solution:**

**Given data:**

The equation is \[{S_n} = 4n - {n^2}\].

Substitute \[n = 1\] in \[{S_n} = 4n - {n^2}\].

${S_1} = 4\left( 1 \right) - {\left( 1 \right)^2}\\

= 4 - 1\\$

= 3

Hence, the first term of Arithmetic progression is 3. The sum of the first term will be the first term that is \[{a_1} = 3\].

Now, substitute \[n = 2\] in the expression \[{S_n} = 4n - {n^2}\].

${S_2} = 4\left( 2 \right) - {\left( 2 \right)^2}\\

= 8 - 4\\

= 4$

Hence, the sum of the first two terms of Arithmetic progression is 4.

Now, calculate the second term of the Arithmetic progression in the following way.

${S_2} = a + {a_2}\\

4 = 3 + {a_2}\\

{a_2} = 1

$

Hence, the second term is \[{a_2} = 1\].

Now, calculate the ${3^{rd}}, {10^{th}}$ and the ${n^{th}}$ term. So, we know about the Arithmetic progression sequence for the ${n^{th}}$ terms is:

\[{a_n} = {a_1} + \left( {n - 1} \right)d\]

Now, calculate the value of ${a_3}$ by substituting the value of ${a_1} = 3,d = - 2\left( {1 - 3} \right),{\rm{ and }}$ n = 3 in \[{a_n} = {a_1} + \left( {n - 1} \right)d\].

${a_3} = 3 + \left( {3 - 1} \right)\left( { - 2} \right)\\

= 3 - 4\\

= - 1$

Hence, the third term is \[{a_3} = - 1\].

Now, calculate the value of ${a_{10}}$ by substituting the value of ${a_1} = 3,d = - 2\left( {1 - 3} \right),{\rm{ and }}$ n = 10 in \[{a_n} = {a_1} + \left( {n - 1} \right)d\].

${a_{10}} = 3 + \left( {10 - 1} \right)\left( { - 2} \right)\\

= 3 - 18\\

= - 15$

Hence, the ${10^{th}}$ term is \[{a_{10}} = - 15\]

Now, to calculate the value of ${a_n}$, substitute the values ${a_1} = 3,\;d = - 2\left( {1 - 3} \right)\;{\rm{and }}$ n in the expression \[{a_n} = {a_1} + \left( {n - 1} \right)d\].

${a_n} = 3 + \left( {n - 1} \right)\left( { - 2} \right)\\

= 3 - 2n + 2\\

= 5 - 2n$

**Hence, the nth term is \[{a_n} = 5 - 2n\].**

**Note:**Make sure do not use the formula of the sum of n terms in Arithmetic progression that is \[{S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\] where a is initial term of the AP and d is the common difference of successive numbers.

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