For four given numbers, the sum of the first two numbers is p, the sum of the second and third number is q and the sum of the last two numbers is r. Find an expression in terms of p, q and r for the sum of the first and last numbers. Also, explain the steps.

s + r - l

Step-by-step explanation:

This is an arithmetic progression question.

The formula to get the terms of an AP is;

L_n = a + (n - 1)d

Where;

a is first term

n is number of term

d is difference between terms

Thus, second term is;

L_2 = a + d

Third term is;

L_3 = a + (3 - 1)d

L_3 = a + 2d

Similarly, L_4 = a + 3d

Now we are told that;

Sum of the first 2 numbers is s;

Sum of the second and third number is l

Sum of the last two numbers is r

Thus;

a + (a + d) = s - - - (eq 1)

(a + d) + (a + 2d) = l - - - (eq 2)

(a + 2d) + (a + 3d) = r - - - (eq 3)

Thus, sum of the first and last term will be;

L_1 + L_4 = a + (a + 3d)

Now,from earlier;

(a + 2d) = r - (a + 3d)

Also;

(a + d) = s - a

Thus, plugging them into the eq 2 gives;

(s - a) + r - (a + 3d) = l

Expanding we have;

s + r - 2a - 3d = l

d = (s + r - 2a - l)/3

Thus;

L_1 + L_4 = a + (a + (3(s + r - 2a - l)/3))

= 2a + s + r - 2a - l = s + r - l

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