, **Title:** A Level Maths | Livestream | Paper 2 | Wed 10th June 2026 | 9:00 PM
**Author:** MathsWithDan
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**1. Sequence and Summation (Iteration)**
* **Problem:** A sequence is defined by `a_n = cos^2(nπ/3)`. Find the exact values of `a1`, `a2`,
`a3` and the sum of `n + cos^2(nπ/3)` from `n=1` to `50`.
* **Key Concepts:** Trigonometric functions, periodic sequences, arithmetic series, sigma notation.
* **Steps & Solutions:**
* **Finding `a1`, `a2`, `a3`:**
* Ensure calculator is in **radians**.
* `a1 = cos^2(π/3) = (1/2)^2 = 1/4`
* `a2 = cos^2(2π/3) = (-1/2)^2 = 1/4`
* `a3 = cos^2(3π/3) = cos^2(π) = (-1)^2 = 1`
* **Identifying Sequence Pattern:**
* Checking further terms (`a4`, `a5`, `a6`) reveals a repeating pattern: `1/4, 1/4, 1, 1/4, 1/4, 1, ...`
* The sequence is periodic.
* **Summation `∑(n + cos^2(nπ/3))` from n=1 to 50:**
* Split the sum: `∑n` + `∑cos^2(nπ/3)`.
* **Sum of `n` (Arithmetic Series):** `1 + 2 + ... + 50`
* Formula: `Sn = n/2 * (2a + (n-1)d)`
* `S50 = 50/2 * (2*1 + (50-1)*1) = 25 * 51 = 1275`.
* **Sum of `cos^2(nπ/3)`:**
* The sum of one cycle (`1/4 + 1/4 + 1`) is `1.5`.
* `50` terms consist of `16` full cycles (`16 * 3 = 48` terms) and `2` remaining terms.
* Sum of 16 cycles: `16 * 1.5 = 24`.
* Sum of the remaining 2 terms (`1/4 + 1/4`): `0.5`.
* Total sum for this part: `24 + 0.5 = 24.5`.
* **Total Sum:** `1275 + 24.5 = 1299.5`.
**2. Small Angle Approximations**
* **Problem:** Show that `cos(θ) + sin(θ/3) - 3*tan(θ) = 5/4` can be approximated as `6θ^2 + 32θ + 3
≈ 0` for small `θ` in radians, and comment on the validity of its solutions.
* **Key Concepts:** Small angle approximations for `sin(θ)`, `cos(θ)`, `tan(θ)`, algebraic
manipulation.
* **Approximations Used:**
* `sin(θ) ≈ θ`
* `cos(θ) ≈ 1 - θ^2/2`
* `tan(θ) ≈ θ`
* **Steps & Solutions:**
**Author:** MathsWithDan
---
**1. Sequence and Summation (Iteration)**
* **Problem:** A sequence is defined by `a_n = cos^2(nπ/3)`. Find the exact values of `a1`, `a2`,
`a3` and the sum of `n + cos^2(nπ/3)` from `n=1` to `50`.
* **Key Concepts:** Trigonometric functions, periodic sequences, arithmetic series, sigma notation.
* **Steps & Solutions:**
* **Finding `a1`, `a2`, `a3`:**
* Ensure calculator is in **radians**.
* `a1 = cos^2(π/3) = (1/2)^2 = 1/4`
* `a2 = cos^2(2π/3) = (-1/2)^2 = 1/4`
* `a3 = cos^2(3π/3) = cos^2(π) = (-1)^2 = 1`
* **Identifying Sequence Pattern:**
* Checking further terms (`a4`, `a5`, `a6`) reveals a repeating pattern: `1/4, 1/4, 1, 1/4, 1/4, 1, ...`
* The sequence is periodic.
* **Summation `∑(n + cos^2(nπ/3))` from n=1 to 50:**
* Split the sum: `∑n` + `∑cos^2(nπ/3)`.
* **Sum of `n` (Arithmetic Series):** `1 + 2 + ... + 50`
* Formula: `Sn = n/2 * (2a + (n-1)d)`
* `S50 = 50/2 * (2*1 + (50-1)*1) = 25 * 51 = 1275`.
* **Sum of `cos^2(nπ/3)`:**
* The sum of one cycle (`1/4 + 1/4 + 1`) is `1.5`.
* `50` terms consist of `16` full cycles (`16 * 3 = 48` terms) and `2` remaining terms.
* Sum of 16 cycles: `16 * 1.5 = 24`.
* Sum of the remaining 2 terms (`1/4 + 1/4`): `0.5`.
* Total sum for this part: `24 + 0.5 = 24.5`.
* **Total Sum:** `1275 + 24.5 = 1299.5`.
**2. Small Angle Approximations**
* **Problem:** Show that `cos(θ) + sin(θ/3) - 3*tan(θ) = 5/4` can be approximated as `6θ^2 + 32θ + 3
≈ 0` for small `θ` in radians, and comment on the validity of its solutions.
* **Key Concepts:** Small angle approximations for `sin(θ)`, `cos(θ)`, `tan(θ)`, algebraic
manipulation.
* **Approximations Used:**
* `sin(θ) ≈ θ`
* `cos(θ) ≈ 1 - θ^2/2`
* `tan(θ) ≈ θ`
* **Steps & Solutions:**
