Consider an arithmetic series and a geometric series having four initial terms from the set \[\left\{ {{\mathbf{11}},{\mathbf{8}},{\mathbf{21}},{\mathbf{16}},{\mathbf{26}},{\mathbf{32}},{\mathbf{4}}} \right\}\]. If the last terms of these series are the maximum possible four-digit numbers, then the number of common terms in these two series is equal to ____

Hint: In this question, we need to find the number of common terms in these two series. For this, we need to write the two series such as arithmetic and geometric series separately. And by observing we can write the number of common terms in these two series.

Complete step-by-step solution:
We have been given two series such as arithmetic series and geometric series.
Let us write these two series separately. in these two series
Thus, observing the arithmetic series given by
Arithmetic series: 11, 16, 21, 26, ….
Also, the geometric series is given by
Geometric series: 4, 8, 16, 32, ….
According to given condition, the terms of geometric series are: 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192
By observing we can say that, all the terms of arithmetic series have unit’s digit either 1 or 6. So in geometric series, there are only three terms which have unit’s digit either 1 or 6.
The common terms in the above two series are 16, 256, and 4096.

Therefore, the common terms in given arithmetic and geometric series are 3.

Additional information: We can say that the arithmetic Progression (AP) is defined as a number series wherein the difference between any two consecutive numbers is a fixed value whereas the Geometric Progression (GP) is defined as a form of series in mathematics wherein every succeeding term is generated by multiplying each preceding term by a fixed number known as a common ratio. This progression is also widely recognized as a pattern-following geometric sequence of numbers.

Note: Many students generally make mistakes in writing the two series such as arithmetic series and geometric series. If so, they may mistakes in finding the common terms between the two series.