Substitute n = 2 into the formula S_n = 4n² - 3n to obtain the sum of the first 2 terms. Calculating this gives S_2 = 10.

if the sum of the first n terms of an ap is Si is equal for square - 3n find its first term common difference and 20th term

Answer: First term $a = 1$ , common difference $d = 8$ , and $20\text{th}$ term $a_{20} = 153$ .

Step-by-step solution

Step 1: Find the first term

The sum of the first term $S_1$ is simply the first term $a_1 = a$ itself. Substituting $n = 1$ into the given formula $S_n = 4n^2 - 3n$ gives $a = 1$ .

Step 2: Find the sum of the first two terms

Substitute $n = 2$ into the formula $S_n = 4n^2 - 3n$ to obtain the sum of the first $2$ terms. Calculating this gives $S_2 = 10$ .

Step 3: Find the second term and common difference

The second term is found by subtracting $S_1$ from $S_2$ , giving $a_2 = 10 - 1 = 9$ . The common difference $d$ is the difference between two consecutive terms, $a_2 - a_1 = 9 - 1 = 8$ .

Step 4: Find the 20th term

Using the $n\text{th}$ term formula $a_n = a + (n - 1)d$ , substitute $a = 1$ , $d = 8$ , and $n = 20$ . This gives $a_{20} = 1 + 19 \times 8 = 153$ .

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