37，40，45，53，66，87，( )。 A. 117 B. 121 C. 128 D. 133

To solve the sequence 37, 40, 45, 53, 66, 87, ( ), we analyze the differences between consecutive terms and identify nested patterns:

Step 1: Calculate first-level differences

Compute the differences between adjacent terms in the original sequence:

\(40 - 37 = 3\)

\(45 - 40 = 5\)

\(53 - 45 = 8\)

\(66 - 53 = 13\)

\(87 - 66 = 21\)

First-level differences: \(3, 5, 8, 13, 21\)

Step 2: Identify the second-level pattern

Calculate the differences of the first-level differences (second-level differences):

\(5 - 3 = 2\)

\(8 - 5 = 3\)

\(13 - 8 = 5\)

\(21 - 13 = 8\)

Second-level differences: \(2, 3, 5, 8\). This is a Fibonacci-like sequence where each term equals the sum of the two preceding terms: \(2+3=5\), \(3+5=8\).

Step 3: Predict the next terms

Following the Fibonacci pattern, the next second-level difference is \(5+8=13\). Adding this to the last first-level difference (\(21\)) gives the next first-level difference: \(21+13=34\).

Step 4: Find the missing term

Add the new first-level difference to the last term in the original sequence: \(87 + 34 = 121\).

Answer: B. 121