32. Arithmetic Progression Problems Part 2#

  1. Let \(S_n\) be the sum of the first \(n\) terms of an arithmetic progression. Then prove that \(S_{n + 3} - 3S_{n + 2} + 3S_{n + 1}- S_n = 0.\)

  2. Let \(a_1, a_2, a_3, ..., a_n, a_{n + 1}, ...\) be an arithmetic progression. Prove that the sequence \(S_1, S_2, S_3, ...\) where \(S_1 = a_1 + a_2 + ... + a_n, S_2= a_{n + 1} + ... + a_{2n}, S_3 = a_{2n + 1} + ... + a_{3n}, ...\) is an arithmetic progression as well as whose common difference is \(n^2\) times higher than the common difference of given progression.

  3. Prove that in any arithmetic progression, whose common difference is not equal to zero, the product of two terms equidistant from the extreme terms is the greater the closer these terms are to the middle term.

  4. Let \(S_k\) be the sum of the first k terms of an arithmetic progression. What must this progression be for the ratio \(\frac{S_{kx}}{S_x}\) to be independent of \(x\)?

  5. Given that \(a_1, a_2, ..., a_n\) form an arithmetic progression. Find the following sum

    \[S = \sum_{i = 1}^{i = n}\frac{a_ia_{i + 1}a_{i + 2}}{a_i + a_{i + 2}}. \]

  6. Let \(a_1, a_2, ..., a_n\) be an arithmetic progression. Prove the identity

    \[\frac{1}{a_1a_n} + \frac{1}{a_2a_{n - 1}} + ... + \frac{1}{a_na_1} = \frac{2}{a_1 + a_n}\left(\frac{1}{a_1} + \frac{1}{a_2} + ... + \frac{1}{a_n}\right) \]

  7. Let \(x_1, x_2, ..., x_n\) form an arithmetic progression. It is known that

    \[x_1 + x_2 + ... + x_n = a, ~~~ x_1^2 + x_2^2 + ... + x_n^2 = b^2 \]

    Determine the progression.

  8. Find the sum

    \[tan^{-1}\frac{r}{1 + a_1a_2} + tan^{-1}\frac{r}{1 + a_2a_3} + ... + tan^{-1}\frac{r}{1 + a_na_{n + 1}} \]

    if \(a_1, a_2, ...\) form an arithmetic progression with a commons difference \(r(a_1>0, r> 0)\).

  9. If \(a_1, a_2, ..., a_n\) are in A. P. with common difference \(d\) then find the sum of the series

    \[\sin d\left[cosec a_1~cosec a_2 + cosec~a_2~cosec~a_3 + ... + cosec~ a_{n - 1}~cosec~a_n\right]. \]

  10. If \(a_1, a_2, ..., a_n\) be an A. P. of positive terms, then prove that

    \[\frac{a_1 + a_{2n}}{\sqrt{a_1} + \sqrt{a_2}} + \frac{a_2 + a_{2n - 1}}{\sqrt{a_2} + \sqrt{a_3}} + ... + \frac{a_n + a_{n + 1}}{\sqrt{a_n} + \sqrt{a_{n + 1}}} = \frac{n(a_1 + a_{2n})}{\sqrt{a_1} + \sqrt{a_{n + 1}}} \]

  11. Find our the value of

    \[\sum_{i = 1}^n\left(\frac{a^i}{b^{i - 1}}\right). \]