Find the 50th term of the sequence 5, 2, 9, 16

To find the 50th term of the sequence, we first need to identify the pattern. The sequence given is: 5, 2, 9, 16.

Let’s look at the differences between the consecutive terms:

• 2 – 5 = -3
• 9 – 2 = 7
• 16 – 9 = 7

It seems complicated at first, but if we observe closely, the sequence can be broken down into a few parts. Notice how the second term (2) drops down from the first term (5) significantly, and then the terms seem to increase by 7 in the subsequent steps.

It helps to also look at the position of the terms. If we label the sequence as:

• a₁ = 5
• a₂ = 2
• a₃ = 9
• a₄ = 16

The odd-indexed terms (a₁, a₃, …) seem to follow one path while the even-indexed terms (a₂, a₄, …) follow another pattern. This leads us to consider two sequences:

• Odd indices: 5, 9, 13, … (which can be simplified to a pattern of +4)
• Even indices: 2, 16, …

The first term for odd-indexed terms is 5 and the increments are consistently 4. Thus, the formula for odd terms is:

a_n = 5 + (n-1) × 4, for n = 1, 3, 5,…

For the 50th term, since 50 is even, we use the odd term formula:

a49 = 5 + (49-1) × 4 = 5 + 48 × 4 = 5 + 192 = 197

So, the 50th term, which corresponds to the 25th odd term, is:

a50 = 197

Therefore, the 50th term of the sequence is 197.