The sum of the first 15 terms in an arithmetic progression is 200, while the sum of the next 15 terms is 350, then the common difference is

Let the first term of A.P. be $$a$$, common difference be $$d$$

So, sum of first n term is $$S_n=\dfrac{n}{2}\left[2a+\left(n-1\right)d\right]$$

Now, using the information given in question,

$$\dfrac{15}{2}\left[2a+14d\right]=200$$

or, $$2a+14d=200\times\ \dfrac{2}{15}=\dfrac{80}{3}$$ ------>(1)

Also, sum of next 15 terms is 350

So, sum of first 30 terms =200+350=550

So, $$\dfrac{30}{2}\left[2a+29d\right]=550$$

or, $$2a+29d=550\times\ \dfrac{2}{30}=\dfrac{110}{3}$$ ------>(2)

Subtracting equation (1) from equation (2),

$$15d=\dfrac{110}{3}-\dfrac{80}{3}$$

or, $$15d=\dfrac{110-80}{3}=\dfrac{30}{3}=10$$

or, $$d=\dfrac{10}{15}=\dfrac{2}{3}$$