If $7^{th}$ term of an arithmetic progression is 28 and $15^{th}$ term is 42, then its common difference will be:-

Question

Accepted Answer

The correct option is 1:
To solve this problem, we need to recall the formula for the $n^{th}$ term of an arithmetic progression (AP). The $n^{th}$ term of an AP is given by the formula:
$$a\_n = a + (n-1)d$$
where $a\_n$ is the $n^{th}$ term, $a$ is the first term, and $d$ is the common difference.

We are given two pieces of information:1.  The $7^{th}$ term of the arithmetic progression is 28.
    So, $a\_7 = 28$. Using the formula, we can write this as:
    $$a + (7-1)d = 28$$
    $$a + 6d = 28 \quad \text{(Equation 1)}$$

2.  The $15^{th}$ term of the arithmetic progression is 42.
    So, $a\_{15} = 42$. Using the formula, we can write this as:
    $$a + (15-1)d = 42$$
    $$a + 14d = 42 \quad \text{(Equation 2)}$$Now we have a system of two linear equations with two variables ($a$ and $d$):1.  $a + 6d = 28$
2.  $a + 14d = 42$To find the common difference $d$, we can subtract Equation 1 from Equation 2:
$$(a + 14d) - (a + 6d) = 42 - 28$$
$$a + 14d - a - 6d = 14$$
$$8d = 14$$
Now, we solve for $d$:
$$d = \frac{14}{8}$$
Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
$$d = \frac{14 \div 2}{8 \div 2}$$
$$d = \frac{7}{4}$$
Thus, the common difference is $\frac{7}{4}$.

The incorrect option is 2: This option would be obtained if there was an error in calculation, possibly by inverting the fraction.
The incorrect option is 3: This option indicates a significant calculation error during the subtraction or division process.
The incorrect option is 4: This option also suggests a calculation mistake, likely a different inversion or incorrect simplification.