Class 11

Math

Algebra

Sequences and Series

An ant moved for several seconds and covered $3$ mm in the first second and $4$ mm more in each successive second than in its predecessor.If the any had covered $1$ mm in first second and $8$ m more in each successive second, then the difference between the path it would cover during the same time and the actual path would be more than $6$ mm but less than $30$ mm. Find the time for which the ant moved (in seconds)

- $5s$
- $4s$
- $6s$
- None of these

Let ant move for t second

Case I: $S_{t}$ when ant move 3 m min first second and 4 m min in each second.

$S_{n}=2t [2(3)+(t−1)4]$

Case II: $S_{t}$ when ant move 1 mm in first second and 8 mm in each second

$S_{t}=2t [2(v)+(t−1)8]$

ATQ

$6<2t [2+(t−1)8]−2t [6+(t−1)4]<30$

$6<2t [−8+4t]<30$

$6<[−4t+2t_{2}]<30$

From option putting t=4 Only condition is satisfied

$6<−16+32<30$

$6<16<30$