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Type the next number in this sequence:\newline00, 11, 33, 66, 1010, 1515, 2121, _\_

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Full solution

Q. Type the next number in this sequence:\newline00, 11, 33, 66, 1010, 1515, 2121, _\_
  1. Identify Pattern: Identify the pattern in the sequence.\newlineEach number appears to be the sum of all natural numbers up to a certain point, which is known as the triangular number sequence.\newlineCalculation: 0=00 = 0, 1=11 = 1, 3=1+23 = 1+2, 6=1+2+36 = 1+2+3, 10=1+2+3+410 = 1+2+3+4, 15=1+2+3+4+515 = 1+2+3+4+5, 21=1+2+3+4+5+621 = 1+2+3+4+5+6.
  2. Determine Next Position: Determine the next position in the sequence.\newlineThe sequence's positions are increasing by 11 each time: 11st, 22nd, 33rd, 44th, 55th, 66th, 77th, ...\newlineThe next position is the 88th.
  3. Calculate 88th Triangular Number: Calculate the 88th triangular number.\newlineUsing the formula for the nth triangular number, Tn=n(n+1)2T_n = \frac{n(n+1)}{2}, where nn is the position in the sequence.\newlineCalculation: T8=8(8+1)2=8×92=36T_8 = \frac{8(8+1)}{2} = \frac{8\times9}{2} = 36.

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