Question

The formula for determining the finite sum, S, of an arithmetic sequence of numbers is S=n/2(a+b), where n is the number of terms, and b is the last term. Express b in terms of a, S, and n.

Answer 1

SOLUTION

GIVEN

The formula for determining the finite sum, S, of an arithmetic sequence of numbers is

$\displaystyle \sf{ S = \frac{n}{2} \big(a + b \big)\: }$

where n is the number of terms, and b is the last term.

TO EXPRESS

b in terms of a, S, and n

EVALUATION

Here it is given that

$\displaystyle \sf{ S = \frac{n}{2} \big(a + b \big)\: }$

where n is the number of terms, and b is the last term

Therefore

$\displaystyle \sf{ S = \frac{n}{2} \big(a + b \big)\: }$

$\implies \displaystyle \sf{ 2S = n \big(a + b \big)\: }$

$\implies \displaystyle \sf{ \frac{2S}{n} = \big(a + b \big)\: }$

$\implies \displaystyle \sf{ \frac{2S}{n} - a= b\: }$

$\implies \displaystyle \sf{ b = \frac{2S}{n} - a\: }$

FINAL ANSWER

$\boxed{ \: \displaystyle \sf{ b = \frac{2S}{n} - a\: } \: }$

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