Sequences & Series

Step-by-step solution

1. 1

   The sequence

   Arithmetic sequence with first term a₁ = 3 and common difference d = 5/2: each term is the previous term plus d.

   $$a_n = a_1 + (n - 1) \cdot d$$
2. 2

   First terms

   Apply a_k = a_1 + (k − 1)·d for k = 1, …, 5:

   • a₁ = 3
   • a₂ = 11/2
   • a₃ = 8
   • a₄ = 21/2
   • a₅ = 13
3. 3

   The n-th term

   Substitute n = 10 into the formula:

   $$a_{10} = 3 + (10 - 1) \cdot \frac{5}{2} = \frac{51}{2}$$
4. 4

   Partial sum

   Add the first 10 terms. Pairing the first with the last term (Gauss' trick) shows the sum is the number of terms times the average of the first and last term.

   $$\begin{aligned} S_n &= \frac{n(a_1 + a_n)}{2} \\ S_{10} &= \frac{10 \cdot (3 + \frac{51}{2})}{2} = \frac{285}{2} \end{aligned}$$
5. 5

   Result

   a₁₀ = 51/2 and S₁₀ = 285/2 — both exact fractions.

   $$a_{10} = \frac{51}{2}, \qquad S_{10} = \frac{285}{2}$$