Which of the following is the general form of the arithmetic progression with the first term as a and the common difference as d

a, a - d, a + 2 d, a - 3 d, . . .

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a, a + d, a + 2 d, a + 3 d, . . .

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a, a - d, a - 2 d, a - 3 d, . . .

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a, a + 2 d, a + 4 d, a + 6 d, . . .

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Solution

The correct option is B $a, a + d, a + 2 d, a + 3 d, . . .$
The general form of an arithmetic progression is given by $a, a + d, a + 2 d, a + 3 d, . . .$ where a is the first term and d is the common difference. Thus the correct answer is ' $b$ '.

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∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{1}{a} & \frac{1}{a (a + d)} & \frac{1}{(a + d) (a + 2 d)} \frac{1}{(a + d)} & \frac{1}{(a + d) (a + 2 d)} & \frac{1}{(a + 2 d) (a + 3 d)} \frac{1}{(a + 2 d)} & \frac{1}{(a + 2 d) (a + 3 d)} & \frac{1}{(a + 3 d) (a + 4 d)} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ \begin{matrix} (a + d) (a + 2 d) & a + 2 d & a 2 d (a + 2 d) & d & d 2 d (a + 3 d) & d & d \end{matrix} ∣ ∣ ∣ ∣ ∣

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∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{1}{a} & \frac{1}{a (a + d)} & \frac{1}{(a + d) (a + 2 d)} \frac{1}{a (a + d)} & \frac{1}{(a + d) (a + 2 d)} & \frac{1}{(a + 2 d) (a + 3 d)} \frac{1}{(a + 2 d)} & \frac{1}{(a + 2 d) (a + 3 d)} & \frac{1}{(a + 3 d) (a + 4 d)} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

\begin{matrix} N o . o f t e r m s & T e r m s & C o m m o n d i f f e r e n c e 3 & a - d, a, a + d & d 4 & a - 3 d, a - d, a + d, a + 3 d & 2 d 5 & a - 2 d, a - d, a, a + d, a + 2 d & d 6 & a - 5 d, a - 3 d, a - d, a + d a + 3 d, a + 5 d & 2 d \end{matrix}

Identify the common difference of the given AP: a, a+d, a+2d, a+3d, ...

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