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?

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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





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


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 =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 =v

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



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

So, p+15=35


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.

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