CBSE Class 10 Maths Chapter 5 Arithmetic Progressions NCERT Solutions

We are given the common difference $d = -4$, the number of terms $n = 7$, and the nth term $a_n = 4$. We need to find the first term $a$. The formula for the nth term of an Arithmetic Progression is:

$$a\_n\; =\; a\; +\; (n-1)d$$

Substitute the given values into the formula:

$4\; =\; a\; +\; (7\; -\; 1)(-4)$

Simplify the equation:

$4\; =\; a\; +\; (6)(-4)$

$4\; =\; a\; -\; 24$

To find $a$, rearrange the equation:

$a\; =\; 4\; +\; 24$

$a\; =\; 28$

Therefore, the first term $a$ is 28. The correct option is C.

We are given the first term $a = 3.5$, the common difference $d = 0$, and the number of terms $n = 101$. We need to find the nth term $a_n$. The formula for the nth term of an Arithmetic Progression is:

$$a\_n\; =\; a\; +\; (n-1)d$$

Substitute the given values into the formula:

$a\_n\; =\; 3.5\; +\; (101\; -\; 1)(0)$

Simplify the equation:

$a\_n\; =\; 3.5\; +\; (100)(0)$

$a\_n\; =\; 3.5\; +\; 0$

$a\_n\; =\; 3.5$

Since the common difference is 0, the AP is a constant sequence, and every term is equal to the first term. Thus, the nth term $a_n$ is 3.5. The correct option is B.

To determine if the given list of numbers is an Arithmetic Progression, we need to check if the difference between consecutive terms is constant.

The given sequence is: $-10, -6, -2, 2, \dots$

Let's denote the terms as $a_1, a_2, a_3, a_4, \dots$

So, $a_1 = -10$, $a_2 = -6$, $a_3 = -2$, $a_4 = 2$.

Calculate the difference between the second term and the first term:

$a\_2\; -\; a\_1\; =\; -6\; -\; (-10)\; =\; -6\; +\; 10\; =\; 4$

Calculate the difference between the third term and the second term:

$a\_3\; -\; a\_2\; =\; -2\; -\; (-6)\; =\; -2\; +\; 6\; =\; 4$

Calculate the difference between the fourth term and the third term:

$a\_4\; -\; a\_3\; =\; 2\; -\; (-2)\; =\; 2\; +\; 2\; =\; 4$

Since the difference between consecutive terms is constant and equal to 4 ($a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = 4$), the given list of numbers is an Arithmetic Progression with a common difference $d = 4$. The correct option is B.

The given Arithmetic Progression is $-5, \frac{-5}{2}, 0, \frac{5}{2}, \dots$

The first term is $a = -5$.

To find the common difference $d$, we subtract the first term from the second term:

$d\; =\; a\_2\; -\; a\_1\; =\; \backslash frac\{-5\}\{2\}\; -\; (-5)\; =\; \backslash frac\{-5\}\{2\}\; +\; 5\; =\; \backslash frac\{-5\; +\; 10\}\{2\}\; =\; \backslash frac\{5\}\{2\}$

We need to find the 11th term, so $n = 11$.

The formula for the nth term of an AP is:

$$a\_n\; =\; a\; +\; (n-1)d$$

Substitute the values of $a$, $n$, and $d$ into the formula:

$a\_\{11\}\; =\; -5\; +\; (11\; -\; 1)\backslash left(\backslash frac\{5\}\{2\}\backslash right)$

Simplify the expression:

$a\_\{11\}\; =\; -5\; +\; (10)\backslash left(\backslash frac\{5\}\{2\}\backslash right)$

$a\_\{11\}\; =\; -5\; +\; \backslash frac\{50\}\{2\}$

$a\_\{11\}\; =\; -5\; +\; 25$

$a\_\{11\}\; =\; 20$

Therefore, the 11th term of the AP is 20. The correct option is B.