Find the sum of the first 20 terms of the AP: 5, 8, 11, 14, and so on.

Step 1: Identify the given values

First, we identify the values provided in the problem. The first term ($a$) is 5. The common difference ($d$) is found by subtracting any term from its succeeding term, so $8 - 5 = 3$. We need to find the sum of the first 20 terms, which means $n = 20$.

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

The formula for the sum of the first $n$ terms of an arithmetic progression ($S_n$) is given by $S_n = \frac{n}{2} [2a + (n-1)d]$. This formula allows us to calculate the sum directly using the first term, common difference, and number of terms.

Step 3: Substitute the values into the formula

Now we substitute the identified values into the sum formula. We replace $a$ with 5, $d$ with 3, and $n$ with 20. This prepares the expression for calculation.

Step 4: Perform the calculations

We simplify the expression step-by-step. First, calculate $\frac{20}{2} = 10$. Then, inside the bracket, calculate $2 \times 5 = 10$ and $(20-1) \times 3 = 19 \times 3 = 57$. Add these two results: $10 + 57 = 67$. Finally, multiply by 10 to get the sum: $10 \times 67 = 670$.

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