Given the sum of the first n terms of a sequence is Sn = 3n2 + n, find the formula for the n-th term, Tn.

Question

ScanSolve AI Tutor · Accepted Answer

Step 1: Write the given formula for the sum of the first n terms, S_n. S_n = 3n² + n Step 2: Find the formula for the sum of the first (n−1) terms, Sn−1, by replacing n with (n−1) in the S_n formula. Sn−1 = 3(n−1)² + (n−1) Expand the term (n−1)²: Sn−1 = 3(n² − 2n + 1) + (n−1) Distribute the 3: Sn−1 = 3n² − 6n + 3 + n − 1 Combine like terms: Sn−1 = 3n² − 5n + 2; Step 3: Use the relationship T_n = S_n − Sn−1 to find the n-th term, T_n. T_n = (3n² + n) − (3n² − 5n + 2) Step 4: Simplify the expression. T_n = 3n² + n − 3n² + 5n − 2; Combine the n² terms: 3n² − 3n² = 0. Combine the n terms: n + 5n = 6n. The constant term is −2. T_n = 6n − 2 The n-th term is: T_n = 6n − 2

Given the sum of the first n terms of a sequence is Sn = 3n2 + n, find the formula for the n-th term, Tn.

Answer

Step-by-step solution

Write the given formula for the sum of the first $n$ terms, $S_n$ .

Find the formula for the sum of the first $(n-1)$ terms, $S_{n-1}$ , by replacing $n$ with $(n-1)$ in the $S_n$ formula.

Use the relationship $T_n = S_n - S_{n-1}$ to find the $n$ -th term, $T_n$ .

Simplify the expression.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Given the sum of the first n terms of a sequence is Sn = 3n2 + n, find the formula for the n-th term, Tn.

Answer

Step-by-step solution

Write the given formula for the sum of the first nnn terms, SnS_nSn​.

Find the formula for the sum of the first (n−1)(n-1)(n−1) terms, Sn−1S_{n-1}Sn−1​, by replacing nnn with (n−1)(n-1)(n−1) in the SnS_nSn​ formula.

Use the relationship Tn=Sn−Sn−1T_n = S_n - S_{n-1}Tn​=Sn​−Sn−1​ to find the nnn-th term, TnT_nTn​.

Simplify the expression.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Write the given formula for the sum of the first $n$ terms, $S_n$ .

Find the formula for the sum of the first $(n-1)$ terms, $S_{n-1}$ , by replacing $n$ with $(n-1)$ in the $S_n$ formula.

Use the relationship $T_n = S_n - S_{n-1}$ to find the $n$ -th term, $T_n$ .