Question 3 of 51beginner🔧 ApplyShort Answer2 marks

Can you write t5t_5, t6t_6, t7t_7 and t8t_8 for the sequence of triangular numbers?

Correct Answer

The sequence of triangular numbers is formed by the sum of the first nn natural numbers. The nthn^{th} term of this sequence is given by the explicit formula tn=n(n+1)2t_n = \frac{n(n+1)}{2}.

Using this formula, we calculate the required terms step-by-step: Step 1: t5=5×62=15t_5 = \frac{5 \times 6}{2} = 15 Step 2: t6=6×72=21t_6 = \frac{6 \times 7}{2} = 21 Step 3: t7=7×82=28t_7 = \frac{7 \times 8}{2} = 28 Step 4: t8=8×92=36t_8 = \frac{8 \times 9}{2} = 36

Therefore, the values are t5=15t_5 = 15, t6=21t_6 = 21, t7=28t_7 = 28, and t8=36t_8 = 36.

Exercise: Exercise | Q: 2 | (Page 3)
For More Understanding

Explanation

The textbook establishes that the sequence of triangular numbers is created by summing the first n natural numbers. It explicitly provides the formula for the nth triangular number as t_n = n(n+1)/2. By substituting n = 5, 6, 7, and 8 into this formula, we arrive at the respective terms of the sequence.

Solution Steps

  1. Step 1: Identify the formula for the nth triangular number: t_n = n(n+1)/2

  2. Step 2: Calculate t_5 = 5(6)/2 = 15

  3. Step 3: Calculate t_6 = 6(7)/2 = 21

  4. Step 4: Calculate t_7 = 7(8)/2 = 28

  5. Step 5: Calculate t_8 = 8(9)/2 = 36

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