How do you write a rule for the nth term of the arithmetic sequence and then find #a_10# for #d=5,

The general form of #n^(th)# term of an AP is given as:-

#a_n = a_1 + (n-1)*d#

In this case, #d = 5 #
#therefore a_n = a_1 + 5*(n-1)#

#a_5 = a_1 + 5*(5-1) = a_1+5*4 = a_1 + 20#

But, #a_5 = 33#

#therefore 33 = a_1 +20 #
#=> a_1 = 33-20 = 13#

Hence #a_n = 13 + 5*(n-1)#

#a_10 = 13 + 5*(10-1) = 13+5*9 = 13+45 = 58#