Number Series

The series is 2, 6, 6, 5, 10, 4, 14, 3, 18, ?. This is two interleaved series. Series 1: 2, 6, 10, 14, 18 (add 4). Series 2: 6, 5, 4, 3, ? (subtract 1). The next term is 3 - 1 = 2.

The differences between consecutive terms are: 15-9=6, 24-15=9, 34-24=10, 51-34=17, 69-51=18, 90-69=21. The pattern of differences should be 6, 9, 12, 15, 18, 21. Replacing 34 with 36 makes the differences 6, 9, 12, 15, 18, 21, which is a consistent pattern.

Pattern: x_n = 2 * x_{n-1} - 5. 9*2-5=13, 13*2-5=21, 21*2-5=37, 37*2-5=69, 69*2-5=133 (not 132).

The series follows a subtraction pattern: 105-20=85, 85-25=60, 60-30=30, 30-35=-5. The term 0 is incorrect; it should be -5.

The series follows a pattern of multiplying the previous term by 2. 15*2=30, 30*2=60, 60*2=120, 120*2=240, 240*2=480.

The series follows the pattern n^2 + 121 (or 11^2, 12^2, 13^2, 14^2, 15^2 plus offsets). Specifically: 122=11^2+1, 170=13^2+1, 226=15^2+1, 290=17^2+1. The next term is 19^2+1 = 361+1 = 362.

The pattern is x * 2 + 2. 6 * 2 + 2 = 14. 14 * 2 + 2 = 30. 30 * 2 + 2 = 62. 62 * 2 + 2 = 126. The number 64 is incorrect; it should be 62.

The series consists of pairs (n^2, n). 49=7^2, 7; 64=8^2, 8; 169=13^2, 13.

The sequence is pairs: (3,6), (24,30), (63,72), (120,132), (195,210). The pattern for the first number in each pair is n^2-1: 2^2-1=3, 5^2-1=24, 8^2-1=63, 11^2-1=120, 14^2-1=195. The second number is n^2+n: 2^2+2=6, 5^2+5=30, 8^2+8=72, 11^2+11=132, 14^2+14=210.

The pattern is: 1*1+1=2, 2*2+1=5, 5*3+1=16, 16*4+1=65, 65*5+1=326, 326*6+1=1957.

This is an arithmetic progression with a=6, d=6. Sum = n/2 * (2a + (n-1)d) = 1800. n/2 * (12 + (n-1)6) = 1800. n * (6 + 3n - 3) = 1800. 3n^2 + 3n - 1800 = 0. n^2 + n - 600 = 0. (n+25)(n-24) = 0. n = 24.

Sum = n/2 * [2a + (n-1)d]. 513 = n/2 * [108 + (n-1)(-3)] = n/2 * [108 - 3n + 3] = n/2 * [111 - 3n]. 1026 = 111n - 3n^2. 3n^2 - 111n + 1026 = 0. n^2 - 37n + 342 = 0. (n-18)(n-19) = 0. Both 18 and 19 work because the 19th term is 54 + 18*(-3) = 0.