Number Series

The sequence follows an arithmetic progression with a common difference of 21 (73+21=94, 94+21=115, 115+21=136, 136+21=157). 147 is incorrect; it should be 157.

The differences between consecutive terms are: 13, 26, 52, 104, 218. The pattern of differences should be 13, 26, 52, 104, 208. The last term 489 is incorrect; it should be 271 + 208 = 479.

t1=2, t2=3. t3=2+3=5, t4=3+5=8, t5=5+8=13.

The general term a_n is found by S_n - S_(n-1). Given S_n = 4n^2 + 3n, then S_(n-1) = 4(n-1)^2 + 3(n-1) = 4(n^2 - 2n + 1) + 3n - 3 = 4n^2 - 8n + 4 + 3n - 3 = 4n^2 - 5n + 1. Subtracting: (4n^2 + 3n) - (4n^2 - 5n + 1) = 8n - 1.

The pattern is a repeating sequence of pairs: (2,2), (3,4), (4,5), (6,6), (7,8), (8,9). The missing number is 5.

Sum = m + 2m + 4m + 8m = 15m. Since m is an integer, the sum must be a multiple of 15. -26 is not a multiple of 15.

The denominators are 2, 6, 12, 20. These are 1*2, 2*3, 3*4, 4*5. The next denominator is 5*6 = 30.

The sequence is 3, 7, 14, 24, 37. Differences are 4, 7, 10, 13. This is an arithmetic progression of differences. The n-th term a_n = a_1 + sum(d_i). Summing the series for 10 terms gives 570.

The differences between consecutive terms are 4, 8, 12. Following this pattern, the next difference should be 16, so 26 + 16 = 42.

Differences: 5-1=4, 11-5=6, 19-11=8, 29-19=10. The differences are increasing by 2. Next difference should be 12. 29+12=41. Check: 41+14=55. Correct.

Sum of n terms Sn = 5n^2 + 2n. First term a1 = S1 = 5(1)^2 + 2(1) = 7. Sum of two terms S2 = 5(2)^2 + 2(2) = 20 + 4 = 24. Second term a2 = S2 - S1 = 24 - 7 = 17.

Let n = 2m. Sum of odd terms = m/2 * (2a + (m-1)2d) = 24. Sum of even terms = m/2 * (2(a+d) + (m-1)2d) = 30. Difference = m*d = 6. Last term - first term = (2m-1)d = 10.5. Since m*d = 6, (2m-1)d = 2md - d = 12 - d = 10.5, so d = 1.5. Then m = 6/1.5 = 4. Number of terms = 2m = 8.