Question 14 of 51intermediate🔧 ApplyNumerical3 marks

Let T1=1T_1 = 1, T2=2T_2 = 2, T3=4T_3 = 4, and Tn=Tn−1+Tn−2+Tn−3T_n = T_{n-1} + T_{n-2} + T_{n-3} for n≥4n \geq 4. Find T4T_4, T5T_5, T6T_6, T7T_7, and T8T_8.

Correct Answer

Given the recursive rule Tn=Tn−1+Tn−2+Tn−3T_n = T_{n-1} + T_{n-2} + T_{n-3} and the initial terms T1=1T_1 = 1, T2=2T_2 = 2, T3=4T_3 = 4, we can find the subsequent terms step-by-step.

Step 1: For n=4n = 4, T4=T3+T2+T1=4+2+1=7T_4 = T_3 + T_2 + T_1 = 4 + 2 + 1 = \mathbf{7}. Step 2: For n=5n = 5, T5=T4+T3+T2=7+4+2=13T_5 = T_4 + T_3 + T_2 = 7 + 4 + 2 = \mathbf{13}. Step 3: For n=6n = 6, T6=T5+T4+T3=13+7+4=24T_6 = T_5 + T_4 + T_3 = 13 + 7 + 4 = \mathbf{24}. Step 4: For n=7n = 7, T7=T6+T5+T4=24+13+7=44T_7 = T_6 + T_5 + T_4 = 24 + 13 + 7 = \mathbf{44}. Step 5: For n=8n = 8, T8=T7+T6+T5=44+24+13=81T_8 = T_7 + T_6 + T_5 = 44 + 24 + 13 = \mathbf{81}.

Therefore, the required terms are T4=7T_4 = 7, T5=13T_5 = 13, T6=24T_6 = 24, T7=44T_7 = 44, and T8=81T_8 = 81.

Exercise: Exercise Set 8.1 | Q: 6 | (Page 7)
For More Understanding

Explanation

The question requires applying a recursive formula to find subsequent terms of a sequence. As explained in the textbook context, a recursive rule describes a sequence by relating terms to previous terms. To use the rule Tn=Tn−1+Tn−2+Tn−3T_n = T_{n-1} + T_{n-2} + T_{n-3}, the earlier terms must be known, and each term must be calculated sequentially since later terms depend on the immediately preceding ones.

Solution Steps

  1. Step 1: Calculate T4 = T3 + T2 + T1 = 4 + 2 + 1 = 7

  2. Step 2: Calculate T5 = T4 + T3 + T2 = 7 + 4 + 2 = 13

  3. Step 3: Calculate T6 = T5 + T4 + T3 = 13 + 7 + 4 = 24

  4. Step 4: Calculate T7 = T6 + T5 + T4 = 24 + 13 + 7 = 44

  5. Step 5: Calculate T8 = T7 + T6 + T5 = 44 + 24 + 13 = 81

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