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Step-by-step explanation:
(a) 1, 4, 9, 16, ...
1², 2², 3², 4², ...
rule : x²
next number : 5² = 25
(b) I think you made a typo here.
the given sequence should be
2, 13, 26, 41, ... correct ?
2 = (1+4)² - 23 = 25 - 23
13 = (2+4)² - 23 = 36 - 23
26 = (3+4)² - 23 = 49 - 23
41 = (4+4)² - 23 = 64 - 23
rule : (x + 4)² - 23
next number : (5+4)² - 23 = 9² - 23 = 81 - 23 = 58
(c) I think you made here a typo too.
the given sequence should be
4, 14, 29, 49, ... correct ?
4 = (5×1² + 5×1 - 2)/2 = (5+5-2)/2 = 8/2
14 = (5×2² + 5×2 - 2)/2 = (20 + 10 - 2)/2 = 28/2
29 = (5×3² + 5×3 - 2)/2 = (45 + 15 - 2)/2 = 58/2
49 = (5×4² + 5×4 - 2)/2 = (80 + 20 - 2)/2 = 98/2
rule : (5×x² + 5×x - 2)/2
next number : (5×5² + 5×5 - 2)/2 = (125 + 25 - 2)/2 =
= 148/2 = 74
(d) 1, 8, 27, 64, ...
1³, 2³, 3³, 4³, ...
rule : x³
next number : 5³ = 125
I could not find any rules for the (b) and (c) as you listed them.
other options for corrections :
2, 13, 26, 51, ...
+11, +13, +15, +17, ...
leading to 68 as next number.
4, 14, 26, 40, ...
+10, +12, +14, +16, ...
leading to 56 as next number.