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Problems Algebra Sequences Progressions Geometric progression Calculus Real numbers Integer and fractional parts. Archimedean property Sums of number sequences and difference series

13–15 4.0

Find the sum $1 / 3 + 2 / 3 + 2^{2} / 3 + 2^{3} / 3 + \dots + 2^{1000} / 3$ .

Problem #PRU-61429

Problems Algebra Sequences Sums of number sequences and difference series

13–16 2.0

Let the sequences of numbers ${a_{n}}$ and ${b_{n}}$ , that are associated with the relation $Δ b_{n} = a_{n}$ ( $n = 1, 2, \dots$ ), be given. How are the partial sums $S_{n}$ of the sequence ${a_{n}}$ $S_{n} = a_{1} + a_{2} + \dots + a_{n}$ linked to the sequence ${b_{n}}$ ?