Which is the (n)th term of the sequence (7,18,33,52,75,\ldots)?
Answer and explanation
Correct answer: (2n^2+5n)
The terms are 7, 18, 33, 52, and 75. Their first differences are 11, 15, 19, and 23, which increase by 4 each time. This indicates a quadratic formula. Check option B, \\(a_n=2n^2+5n\\): at n = 1 it gives \\(2+5=7\\); at n = 2 it gives \\(8+10=18\\); at n = 3 it gives \\(18+15=33\\); and at n = 4 it gives \\(32+20=52\\). It also gives 75 at n = 5, so B is correct.
The constant second difference is 4. For a quadratic expression \\(an^2+bn+c\\), the second difference equals \\(2a\\), so here the coefficient of \\(n^2\\) is 2. Since the first term is 7, the remaining linear part is fixed by the next terms, giving \\(2n^2+5n\\). Option A fails at the first term, while the other choices also fail on direct substitution. The supplied answer B is accurate.
Frequently asked questions
What is the correct answer to this question?
(2n^2+5n)
Why is this the correct answer?
The terms are 7, 18, 33, 52, and 75. Their first differences are 11, 15, 19, and 23, which increase by 4 each time. This indicates a quadratic formula. Check option B, \\(a_n=2n^2+5n\\): at n = 1 it gives \\(2+5=7\\); at n = 2 it gives \\(8+10=18\\); at n = 3 it gives \\(18+15=33\\); and at n = 4 it gives \\(32+20=52\\). It also gives 75 at n = 5, so B is correct.
The constant second difference is 4. For a quadratic expression \\(an^2+bn+c\\), the second difference equals \\(2a\\), so here the coefficient of \\(n^2\\) is 2. Since the first term is 7, the remaining linear part is fixed by the next terms, giving \\(2n^2+5n\\). Option A fails at the first term, while the other choices also fail on direct substitution. The supplied answer B is accurate.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Sequences and Progressions. Topic: nth term.
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