Let's look at the following question.
Although this is a geometric series, it uses a concept called arithmetic average.
Using the properties of geometric series, all terms are represented by a2 and Q, and a2 is removed from both sides of the equal sign, so a quadratic equation with one variable about Q can be obtained.
To solve this equation, since all terms are positive, we can get the final answer by discarding negative values.
Arithmetic sum and ratio, two special series, can be transformed into each other by taking logarithm or exponential power. So sometimes the topic of geometric series will be examined in combination with the nature of logarithmic operation, such as the following question.
Logarithms with the same radix are added, the radix is unchanged, and real numbers are multiplied.
According to the nature of the proportional term, the product of the first five terms is only related to the third term. Finally, combined with logarithmic algorithm, the final answer can be obtained.
Finally, let's look at a question like this, which is the final exam question of Jiangsu Suqian 202 1.
We need to find the general term formula of the sequence {an} according to the known conditions.
Finally, an is changed into an exponential power with the base of 2, which is convenient for us to further observe what to do next.
What we require is the maximum value of the product of the first n terms of the sequence {an}, where an is an exponential power with the base of 2, and multiplied with the base power, the base invariant index is added, and finally it is transformed into the maximum value problem of the sum of the first n terms of arithmetic progression.
How to get this arithmetic progression {bn}? It's very simple. Just take the logarithm of an to base 2.
See how well the friends have mastered the contents of the last issue. Do you still remember the two methods to find the first n terms and the best value of arithmetic series? Here we use the method of quadratic function to find the first n terms and Sn.
Then judging the opening direction and symmetry axis, the maximum value of Sn can be obtained. Note that n is a positive integer.
Finally, let the product of the first n items of the sequence {an} be t n, and the relationship between Tn and Sn can be obtained, so that the maximum value of Tn can be obtained from the maximum value of Sn.