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# The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

Given that,
first term, a = 5
last term, l = 45
Sum of the AP, $$S_n$$ = 400

As we know, the sum of AP formula is;

$$\Rightarrow S_n = {{n} \over {2}} (a+l)$$
$$\Rightarrow$$ 400 = $${{n} \over {2}}$$(5+45)
$$\Rightarrow$$ 400 = $${{n} \over {2}}$$(50)

Number of terms, n =16
As we know, the last term of AP series can be written as;

l = a+(n -1)d
$$\Rightarrow$$ 45 = 5 +(16 -1)d
$$\Rightarrow$$ 40 = 15d
$$\Rightarrow$$ Common difference, d = $${{40} \over {15}} = {{8} \over {3}}$$