Choose the correct choice in the following and justify : (i) 30th term of the AP: , is (A) (B) (C) (D) (ii) 11th term of the AP: , is (A) (B) (C) (D)

Understand the Question The -th term of an Arithmetic Progression (AP) is given by the formula: where is the first term, is the common difference (), and is the term number. To find the specified term, identify , calculate , and substitute them along with into the formula. Part (i) 30th term of the AP: , is (A) (B) (C) (D) Step 1 · Calculate the 30th Term Given AP: First term, Common difference, : For the term, . Using Answer > (i) (C) Part (ii) 11th term of the AP: , is (A) (B) (C) (D) Step 1 · Calculate the 11th Term Given AP: First term, Common difference, : For the term, . Using Answer > (ii) (B) Common Mistakes Sign of Common Difference (): Always calculate . For a decreasing AP like , , not . Handling Double Negatives: When subtracting a negative first term, remember that , e.g., .

Arithmetic Progressions | Exercise 5.2

Question 2

Choose the correct choice in the following and justify :

(i) 30th term of the AP: $10, 7, 4, \dots$ , is (A) $97$ (B) $77$ (C) $-77$ (D) $-87$

(ii) 11th term of the AP: $-3, -\dfrac{1}{2}, 2, \dots$ , is (A) $28$ (B) $22$ (C) $-38$ (D) $-48\dfrac{1}{2}$

Solution

Understand the Question

The $n$ -th term of an Arithmetic Progression (AP) is given by the formula: $a_n = a + (n - 1)d$ where $a$ is the first term, $d$ is the common difference ( $d = a_2 - a_1$ ), and $n$ is the term number.
To find the specified term, identify $a$ , calculate $d$ , and substitute them along with $n$ into the formula.

(i) 30th term of the AP: $10, 7, 4, \dots$ , is (A) $97$ (B) $77$ (C) $-77$ (D) $-87$

Step 1 · Calculate the 30th Term

Given AP: $10, 7, 4, \dots$

First term, $a = 10$

Common difference, $d$ :

\begin{aligned} d &= 7 - 10 \\[0.6em] &= -3 \end{aligned}

For the $30\text{th}$ term, $n = 30$ .

Using $a_n = a + (n - 1)d$

\begin{aligned} a_{30} &= 10 + (30-1)(-3) \\[0.6em] &= 10 + (29)(-3) \\[0.6em] &= 10 - 87 \\[0.6em] &= -77 \end{aligned}

Answer

(i) (C) $-77$

(ii) 11th term of the AP: $-3, -\dfrac{1}{2}, 2, \dots$ , is (A) $28$ (B) $22$ (C) $-38$ (D) $-48\dfrac{1}{2}$

Step 1 · Calculate the 11th Term

Given AP: $-3, -\dfrac{1}{2}, 2, \dots$

First term, $a = -3$

Common difference, $d$ :

\begin{aligned} d &= -\dfrac{1}{2} - (-3) \\[0.6em] &= -\dfrac{1}{2} + 3 \\[0.6em] &= -\dfrac{1}{2} + \dfrac{6}{2} \\[0.6em] &= \dfrac{5}{2} \end{aligned}

For the $11\text{th}$ term, $n = 11$ .

Using $a_n = a + (n - 1)d$

\begin{aligned} a_{11} &= -3 + (11-1)\left(\dfrac{5}{2}\right) \\[0.6em] &= -3 + (10)\left(\dfrac{5}{2}\right) \\[0.6em] &= -3 + 5 \times 5 \\[0.6em] &= -3 + 25 \\[0.6em] &= 22 \end{aligned}

Answer

(ii) (B) $22$

Common Mistakes

Sign of Common Difference ( $d$ ): Always calculate $d = a_2 - a_1$ . For a decreasing AP like $10, 7, 4, \dots$ , $d = 7 - 10 = -3$ , not $+3$ .
Handling Double Negatives: When subtracting a negative first term, remember that $-\left(-\text{number}\right) = +\text{number}$ , e.g., $-\dfrac{1}{2} - (-3) = -\dfrac{1}{2} + 3 = \dfrac{5}{2}$ .

Question 2

More questions in Exercise 5.2