Number Series

For n=1, 2(1)-5 = -3. For n=2, 2(2)-5 = -1. For n=3, 2(3)-5 = 1. The series is -3, -1, 1...

The sequence is n^2 - 1 for n=0, 1, 2, 3, 4, 5. 0^2-1 = -1, 1^2-1 = 0, 2^2-1 = 3, 3^2-1 = 8, 4^2-1 = 15, 5^2-1 = 24. The missing term is 3.

The differences between consecutive terms are: 27-24=3, 31-27=4, 33-31=2, 36-33=3, 39-36=3. The pattern of differences is 3, 3, 3, 3. If the sequence were 24, 27, 30, 33, 36, 39, the differences would be constant. Thus, 31 is the wrong term.

The differences are: 13-8=5, 21-13=8, 32-21=11, 47-32=15, 63-47=16, 83-63=20. The pattern of differences is 5, 8, 11, 14, 17, 20. 47 is incorrect because 32+14=46, and 46+17=63.

The sequence follows the pattern n^2 + 1: 2^2+1=5, 3^2+1=10, 4^2+1=17, 5^2+1=26 (not 24), 6^2+1=37, 7^2+1=50, 8^2+1=65. Thus, 24 is the wrong term.

The sequence follows the pattern 2^n (1, 2, 4, 8, 16, 32, 64). The next term should be 128, not 96.

The pattern is 2^n - 1: 2^1-1=1, 2^2-1=3, 2^3-1=7, 2^4-1=15, 2^5-1=31 (not 27), 2^6-1=63, 2^7-1=127. Thus, 27 is the wrong term.

The series is arithmetic with first term a=25 and common difference d=-4. The sum of n terms is given by S_n = (n/2)(2a + (n-1)d). Plugging in n=16, a=25, and d=-4 gives S_16 = 8(50 + 15(-4)) = 8(50 - 60) = 8(-10) = -80.

The sum of an arithmetic series is given by S = (n/2) * (a1 + an). With n = 20, a1 = 10, and a20 = 100, the sum is (20/2) * (10 + 100) = 10 * 110 = 1100.

The sequence 1, 3, 5, 7 is an arithmetic progression with a common difference of 2. The next two terms are 7+2=9 and 9+2=11.

Arithmetic sequence: a = 120, d = -5, n = 40. Sum = (n/2) * [2a + (n-1)d] = 20 * [240 + 39 * (-5)] = 20 * [240 - 195] = 20 * 45 = 900.

Sum = (n/2) * [2a + (n-1)d]. n = 20, a = -12, d = -5. Sum = (20/2) * [2(-12) + (19)(-5)] = 10 * [-24 - 95] = 10 * [-119] = -1190.

For n=1, 5(1)-4=1. For n=2, 5(2)-4=6. For n=3, 5(3)-4=11. For n=4, 5(4)-4=16. The sequence is 1, 6, 11, 16.