### Analysis of Arithmetic Series; Statements Judging

Question Sample Titled 'Analysis of Arithmetic Series; Statements Judging'

If the sum of the first ${n}$ terms of a sequence is ${4}{n}^{{2}}+{4}{n}$ , which of the following is/are true?

 I. ${18}$ is a term of the sequence. II. The ${1}$st term of the sequence is ${4}$ . III. The sequence is a geometric sequence.

A
None of the above
B
I only
C
I and II only
D
I and III only

 Let ${T}{\left({n}\right)}$ be the general term of the sequence, and ${S}{\left({n}\right)}$ be the sum of the first ${n}$ terms of the sequence.  By the given formula, we have ${S}{\left({1}\right)}$ $={8}$ ${S}{\left({2}\right)}$ $={24}$ ${S}{\left({3}\right)}$ $={48}$  Note that ${T}{\left({1}\right)}$ $={S}{\left({1}\right)}$  $={8}$  ${T}{\left({2}\right)}$ $={S}{\left({2}\right)}-{S}{\left({1}\right)}$  $={16}$  ${T}{\left({3}\right)}$ $={S}{\left({3}\right)}-{S}{\left({2}\right)}$  $={24}$  ∵   ${T}{\left({3}\right)}-{T}{\left({2}\right)}$ $={T}{\left({2}\right)}-{T}{\left({1}\right)}$ ∴   The sequence is a arithmetic sequence.  ${18}$ lies between ${T}{\left({2}\right)}$ and ${T}{\left({3}\right)}$.  ∴   ${18}$ is not a term of the sequence.

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