# Given the positive integers p, q, r, s, t, u, v, if the sum of the values of each group of three consecutive letters is 35, and q+u=15, then what is p+q+r+s+t+u+v?

In this question, we have seven positive integers (p, q, r, s, t, u, and v). It's said that the sum of each group of three consecutive letters equals 35. Also, q+u=15.

So, by inference, we can conclude:

p+q+r= 35

q+r+s=35

r+s+t=35

s+t+u=35

t+u+v=35

So, from the above, we can conclude that p+q+r=q+r+s. Thus:

p+q+r=q+r+s

p =q+r-q-r+s

p =s

From the above, we can also conclude that r+s+t=s+t+u. Thus:

r+s+t=s+t+u

r =s+t-s-t+u

r =u

Also, from the above, we see q+r+s=r+s+t. Thus, q=t.

Finally, we see that that s+t+u=t+u+v. Thus:

s+t+u=t+u+v

s = t+u-t-u+v

s =v

So, to go back to the original equation, we will perform substitutions.

p+q+r=35

q+u=15

Since r=u, we can write p+q+u=35

So, p+15=35

p=20

Since r=u and q+u=15, we can write q+u+s=35

So, s=20

To add all the positive integers, we get:

p+q+r+s+t+u+v= 20+q+r+20+t+u+20

Remember that q+r+s=35. If s=20, then q+r=15

Remember also that t+u+v=35. If v=20 (remember v=s and s=20), then t+u=15

To add it all up, we get 20+15+20+15+20, which equals 90.