Download Some problems on Sequences

Some problems on Sequences
4.011. The sum of three numbers, forming an arithmetic sequence, is 2, and the sum of the squares of
these numbers is 14/9. Find the numbers.
The answer: {1/3, 2/3, 1}.
4.019. Find the first term and the ratio of the geometric sequence if it is known that the difference of
the fourth and the second terms is -45/32 and the difference of the sixth and the fourth term is -45/512.
The answer: {6, 1/4}, {-6, -1/4}.
4.21. It is known that the given arithmetic sequence has the following property: for any given n, the
sum of its n terms is 5𝑛2 . Find the common difference and the first three terms.
The answer: {10, 5, 15, 25}.
4.044. Find four numbers, if the first three form geometric sequence, and the last three form
arithmetic sequence. The sum of the extremes (the first and the last) numbers is 21, and the sum of the
means (two numbers in the middle) is 18.
The answer: {3, 6, 12, 18}, {18.75, 11.25, 6.75, 2.25}.
4.067. The sum of three numbers is 11/18, and the sum of their inverses, forming an arithmetic
sequence, is 18. Find these numbers.
The answer: {1/9, 1/6, 1/3}.