Consider an arithmetic series and a geometric series having four initial terms from the se…

Question

Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these ...

Answer: 3

Accepted Answer

From the set: A.P. $= \{11,16,21,26,\ldots\}$ (common difference $5$), G.P. $=\{4,8,16,32,\ldots\}$ (ratio $2$).

Common terms satisfy $5m+6 = 2^{n+1}$ for integers $m,n$, so $2^{n+1}-6$ must be a multiple of $5$. Checking the last digit of powers of $2$ ($2,4,6,8,\ldots$), this requires $2^{n+1}$ to end in $6$: $2^{n+1} = 16, 256, 4096, 65536,\ldots$

Restricting to four-digit-or-smaller terms consistent with both series' maximum four-digit last terms, the common terms are $\{16, 256, 4096\}$ --- 3 common terms.

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