\(\text{If } T_n = T_{n-1} + T_{n-2} \quad \forall\, n \geq 3 \text{ and } T_1 = T_2 = 1 \text{ then } T_1 + T_2 + \ldots + T_{10} =\ ? \)

Given:

If T_n = T_n-1 + T_n-2 ∀ n ≥ 3 and T₁ = T₂ = 1, then find T₁ + T₂ + ... + T₁₀.

Formula used:

Fibonacci series formula: T_n = T_n-1 + T_n-2

Sum of terms: S = T₁ + T₂ + ... + T₁₀

Calculation:

Calculate Fibonacci terms up to T₁₀

T₁ = 1

T₂ = 1

T₃ = T₂ + T₁ = 1 + 1 = 2

T₄ = T₃ + T₂ = 2 + 1 = 3

T₅ = T₄ + T₃ = 3 + 2 = 5

T₆ = T₅ + T₄ = 5 + 3 = 8

T₇ = T₆ + T₅ = 8 + 5 = 13

T₈ = T₇ + T₆ = 13 + 8 = 21

T₉ = T₈ + T₇ = 21 + 13 = 34

T₁₀ = T₉ + T₈ = 34 + 21 = 55

Calculate the sum S

⇒  S = T₁ + T₂ + T₃ + T₄ + T₅ + T₆ + T₇ + T₈ + T₉ + T₁₀

⇒  S = 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143

Option 2: T₁₂ - 1

⇒  T₁₂ = T₁₁ + T₁₀ = 89 + 55 = 144

⇒  T₁₂ - 1 = 144 - 1 = 143

The Correct answer is Option 2.