the sum of the first 15 terms of an ap is 450 and its first term is 10 find its common difference

Answer: The common difference of the AP is $\frac{20}{7}$ .

Step-by-step solution

Step 1: Identify the given values

From the given problem, the sum of the first $15$ terms is $S_{15} = 450$ , the first term is $a = 10$ , and the number of terms is $n = 15$ . We need to find the common difference, denoted by $d$ .

Step 2: State the formula for the sum of an AP

We use the standard formula for the sum of the first $n$ terms of an Arithmetic Progression when the first term $a$ and common difference $d$ are involved: $S_n = \frac{n}{2} [2a + (n - 1)d]$ .

Step 3: Substitute the given values into the formula

Substitute $S_n = 450$ , $n = 15$ , and $a = 10$ into the sum formula. This sets up a linear equation in one variable, $d$ .

Step 4: Simplify the equation

Inside the brackets, multiply $2 \times 10 = 20$ and subtract $15 - 1 = 14$ . Factoring out $2$ from $[20 + 14d]$ cancels the denominator $2$ , giving $15(10 + 7d)$ .

Step 5: Solve for the common difference d

Divide both sides by $15$ to get $30 = 10 + 7d$ . Subtract $10$ from both sides to find $7d = 20$ . Dividing by $7$ gives $d = \frac{20}{7}$ .

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