Question

The sum of n terms of arithmetic series is ${S_n} = 2n - {n^2}$. Find the first term and the common difference.
A) $a = 2;d = 2$
B) $a = 0;d = 1$
C) $a = 1,d = - 2$
D) $a = - 1,d = 2$

Answer 1

Hint: First, find the first and second terms of the arithmetic series by the formula ${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$ and ${S_n} = 2n - {n^2}$. Then compare the values of the first term obtained by both formulae which will give the first term. After that compare the values of the second term obtained by both formulae. Then, substitute the value of the first term and calculate the common difference.

Formula used:
The sum of n terms of the arithmetic series is,
${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$ …..(1)
where n is the number of terms, a is the first term and d is the common difference.

Complete step-by-step answer:
The sum of n terms of the arithmetic series is,
${S_n} = 2n - {n^2}$ …..(2)
The first term by equation (1) is,
${S_1} = \dfrac{1}{2}\left[ {2a + \left( {1 - 1} \right)d} \right]$
Simplify the equation by doing necessary calculations,
${S_1} = \dfrac{1}{2}\left( {2a} \right)$
Open brackets to cancel out 2 from numerator and denominator,
${S_1} = a$ ……(3)
The second term by equation (1) is,
${S_2} = \dfrac{2}{2}\left[ {2a + \left( {2 - 1} \right)d} \right]$
Simplify the equation by doing necessary calculations,
${S_2} = 2a + d$ ……(4)
The first term by equation (2) is,
\[{S_1} = 2 \times 1 - {\left( 1 \right)^2}\]
Simplify the equation by doing necessary calculations,
${S_1} = 2 - 1$
Subtract 1 from 2 in the RHS,
${S_1} = 1$ ……(5)
The second term by equation (2) is,
${S_2} = 2 \times 2 - {\left( 2 \right)^2}$
Simplify the equation by doing necessary calculations,
${S_2} = 4 - 4$
Subtract 4 from 4 in the RHS,
${S_2} = 0$ ……(6)
As, equation (3) and (5), both contain the value of ${S_1}$. Compare them,
$a = 1$ ……(7)
As, equation (3) and (5), both contain the value of ${S_1}$. Compare them,
$2a + d = 0$
Substitute the value of a from equation (7),
$2 \times 1 + d = 0$
Subtract 2 from both sides,
$2 + d - 2 = 0 - 2$
Simplify the equation by doing necessary calculations,
$d = - 2$
Therefore, the first term is 1 and the common difference is -2.

Hence, option (C) is correct.

Note: The students must remember to use the formula of the sum of the arithmetic series. Many times, the students get confused and use arithmetic sequence formulas instead of the sum of arithmetic series. Also, he/she should take care when performing the addition or subtraction to get the correct result. Otherwise, he/she will make mistakes and it will lead to the wrong answer.