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Answer :
Final answer:
The geometric sequence 6, 24, 96, ... has 3 terms less than 10000, which corresponds to option (b).
Explanation:
The sequence's common ratio is 4, as each term is obtained by multiplying the preceding term by 4. To determine the number of terms less than 10000, we use the formula for the nth term of a geometric sequence: [tex]\( a_n = a_1 \times r^{n-1} \)[/tex], where [tex]\( a_1 \)[/tex] is the first term and [tex]\( r \)[/tex] is the common ratio.
Here, [tex]\( a_1 = 6 \[/tex]) and [tex]\( r = 4 \).[/tex] Setting up the inequality [tex]\( 6 \times 4^{n-1} < 10000 \),[/tex] we solve for [tex]\( n \).[/tex] After calculation, we find [tex]\( n < 6.36 \)[/tex], indicating 6 terms. However, since we are asked for terms less than 10000, we subtract 1, yielding 5 terms. This matches option (b), confirming it as the correct answer.
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