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Step-by-Step Solution
Step 1: Identify the Set and Possible Patterns
The given set is {11, 8, 21, 16, 26, 32, 4}. We are told there is an arithmetic progression (A.P.) and a geometric progression (G.P.) each starting with four initial terms from this set. We also know that these progressions continue until their last terms are the maximum possible four-digit numbers.
Step 2: Determine the Arithmetic Progression (A.P.)
Observing the set, the only way to pick four terms that form an A.P. is:
11, 16, 21, 26
These terms have a common difference:
d = 16 - 11 = 5
So the A.P. is 11, 16, 21, 26, \dots and it can be written in general form as:
a_n = 11 + (n-1)\times 5
This sequence is extended further until it remains a four-digit number (≤ 9999).
Step 3: Determine the Geometric Progression (G.P.)
From the set, four initial terms that form a G.P. are:
4, 8, 16, 32
These terms have a common ratio:
r = \frac{8}{4} = 2
Thus, the G.P. is 4, 8, 16, 32, \dots and in general form:
g_n = 4 \times 2^{(n-1)}
This sequence is similarly continued until it remains a four-digit number (≤ 9999). The largest four-digit term here is 8192 (since 4 \times 2^{12-1} = 4 \times 2048 = 8192 , which is still a four-digit number).
Step 4: Find Common Terms in Both Sequences
First common term:
A.P. terms start 11, 16, 21, 26, …
G.P. terms start 4, 8, 16, 32, …
Clearly, 16 is common to both.
Check next powers of 2 in the G.P. (64, 128, 256, …) against the A.P. formula 11 + 5(n-1) .
256 in the G.P.: We see if 256 is in the A.P. by solving 256 = 11 + 5(n-1) . This gives 245 = 5(n-1) \Rightarrow n-1 = 49 \Rightarrow n = 50 . Hence, 256 is indeed in the A.P.
Check subsequent terms in the G.P. (512, 1024, 2048, 4096, 8192) similarly.
4096 in the G.P.: We see if 4096 is in the A.P. by solving 4096 = 11 + 5(n-1) . This gives 4096 - 11 = 4085 = 5(n-1) \Rightarrow n-1 = 817 \Rightarrow n = 818. Thus, 4096 is also a term of the A.P.
Therefore, the common terms (up to four digits) in both sequences are:
16, 256, and 4096.
Step 5: Conclude the Number of Common Terms
We have exactly three common terms: 16, 256, and 4096.
Answer: 3
