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?

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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.

Last Updated by eNotes Editorial on