The 4 ^(t)h term of GP is 500 and its common ratio is 1/m,m. Let S n denote the sum of the first n terms of this GP. If S₆ > S₅ + 1 and S₇ < S₆ + 1/2, then the number of possible values of m is _ _ _ _ _ _

Sequence and Series: JEE Main 2023 Mathematics question

JEE Main 2023 · 24 Jan, Morning shift · Q45

JEE Main 202324 Jan, Morning shiftMathematicsMediumSequence and Series179× since 2002

The 4

^\mathrm{th}

term of GP is 500 and its common ratio is

\frac{1}{m},m\in\mathbb{N}

. Let

\mathrm{S_n}

denote the sum of the first n terms of this GP. If

\mathrm{S_6 > S_5 + 1}

and

\mathrm{S_7 < S_6 + \frac{1}{2}}

, then the number of possible values of m is ___________

Your answerType your answer. Integers and decimals are accepted.