Number Series

This recursive sequence follows the pattern T_n = T_(n-1) + T_(n-2) + T_(n-3). Each term is the sum of the three preceding terms. Verifying: 2+4+7=13, 4+7+13=24, and 7+13+24=44.

The series follows the pattern: multiply by 2 and then add decreasing numbers (2, 1, 0, -1, -2). Starting from 5: 5×2+2=12, 12×2+1=25, 25×2+0=50, 50×2-1=99, 99×2-2=196. This pattern consistently generates the next term in the sequence.

Let AP1 be: a, a+d, a+2d, ... and AP2 be: b, b+e, b+2e, ... The product sequence is: ab, (a+d)(b+e), (a+2d)(b+2e), ... Term 1: 192 = ab, Term 2: 360 = (a+d)(b+e) = ab + ae + bd + de, Term 3: 576 = (a+2d)(b+2e). From 192 to 360, difference is 168. From 360 to 576, difference is 216. The second difference is 216-168=48. For product of two APs, the 8th term follows the pattern. Through calculation: if we factor 192 = 12×16, 360 = 15×24, 576 = 18×32, then AP1: 12, 15, 18, 21, 24, 27, 30, 33 and AP2: 16, 24, 32, 40, 48, 56, 64, 72. Term 8 = 33 × 72 = 2376.

The pattern is $x_{n} = x_{n-1} \times (x_{n-1} + 1)$. Specifically: $1 \times (1+1) = 2$; $2 \times (2+1) = 6$; $6 \times (6+1) = 42$; $42 \times (42+1) = 1806$. The next term is $1806 \times 1807 = 3263442$. This recursive growth explains the rapid increase in values.

The sequence consists of two interleaved series. Series 1 (odd positions): 3, 7, ?, 9. Series 2 (even positions): 10, 8, 12, 16. Series 1 logic: +4, then -2? No. Series 2 logic: -2, +4, +4. Actually, Series 1 is 3, 7, 11, 15 (adding 4). 11 fits perfectly.

The differences are squares: +1, +4, +9, +16, +25. The next difference should be +36. 56 + 36 = 92. Therefore, 91 is the wrong number in the sequence.

The differences between consecutive numbers are prime numbers: +2, +3, +5, +7, +11. Following 153, the next difference should be +13, making the final number 166. Therefore, 165 is the wrong number.

The pattern is (x * 3) - 4. (10*3)-4=26; (26*3)-4=74; (74*3)-4=218; (218*3)-4=650. The number 654 is wrong because it should be 650 based on the established mathematical progression.

Sequence is cubes: 1³=1, 2³=8, 3³=27, 4³=64, 5³=125, 6³=216, 7³=343. 124 breaks the pattern where 125 should be (5³=125). 124 is incorrect.

Pattern alternates between incrementing and constant: positions 1,3,5,7 are 5,6,7,? (should be 8 by incrementing), while positions 2,4,6 are all 16. Position 7 should be 8, not 9. 9 breaks the alternating pattern.

The series follows the pattern of n^3 + 1. For n=1, 1^3+1=2; for n=2, 2^3+1=9; for n=3, 3^3+1=28; for n=4, 4^3+1=65. The next term for n=5 is 5^3+1, which equals 125+1 = 126.

14, 18 ___, 26, 30

Add 4 to each term to get the next term. So, 14 + 4 = 18 18 + 4 = 22 22 + 4 = 26                    26 + 4 = 30

9, 11, 13, __ Pattern: Add 2 to each term to get the next term. 9 + 2 = 11 11 + 2 = 13 13 + 2 = 15