Find the sum of the first 20 terms of the AP: 5, 8, 11, 14, and so on.
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Step-by-Step Solution
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 (Sn) is given by Sn=2n[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 220=10. Then, inside the bracket, calculate 2×5=10 and (20−1)×3=19×3=57. Add these two results: 10+57=67. Finally, multiply by 10 to get the sum: 10×67=670.