Find the 7th term of the sequence: 2, 5, 10, 17, 26,...

Analyze the question

We need the 7th term of the sequence 2, 5, 10, 17, 26, ...

Find the pattern

Look at the differences between consecutive terms:

• 5 − 2 = 3
• 10 − 5 = 5
• 17 − 10 = 7
• 26 − 17 = 9

The first differences (3, 5, 7, 9) increase by 2 each time, so the sequence is quadratic.

Find the nth-term formula

Compare with square numbers:

• n = 1: 1² + 1 = 2
• n = 2: 2² + 1 = 5
• n = 3: 3² + 1 = 10
• n = 4: 4² + 1 = 17
• n = 5: 5² + 1 = 26

So the nth term is Tₙ = n² + 1.

Find the 7th term

T₇ = 7² + 1 = 49 + 1 = 50

Check using differences: the next differences are 11 and 13.

• 6th term = 26 + 11 = 37
• 7th term = 37 + 13 = 50 ✓

Note that 37 (option A) is the 6th term, which is a common trap. Always check which term the question asks for.

50**.

Tip: When the differences between terms are not constant but increase by a constant amount (the second difference is constant), think of n² plus or minus something. Compare each term to perfect squares to find the rule quickly.