Question Type 1: Finding the sum to infinity of a given sequence Exercises

Calculate the sum to infinity of the geometric sequence with first term $u_1 = 5$ and common ratio $r = \frac{1}{3}$.

Calculate the sum to infinity of the series $3 + \tfrac{3}{2} + \tfrac{3}{4} + \tfrac{3}{8} + \dots$.

Show that the infinite sum of the sequence $7, \tfrac{7}{4}, \tfrac{7}{16}, \dots$ equals $\tfrac{28}{3}$.

Find the sum to infinity of the sequence $8, -4, 2, -1,\dots$.

Find the sum to infinity of the geometric sequence defined by $u_n = 2\cdot(0.8)^{n-1}$.

Determine the sum to infinity of the geometric sequence with first term $u_1=12$ and common ratio $r=\frac{3}{4}$.

Given that the sum to infinity of a geometric series is $20$ and its first term is $5$, find the common ratio $r$.

A geometric series has common ratio $r=\tfrac{2}{5}$ and sum to infinity $15$. Find its first term.

Calculate the sum to infinity of the combined series

$6 + \tfrac{3}{2} + \tfrac{3}{8} + \dots\; -\; \bigl(2 + 1 + \tfrac{1}{2} + \dots\bigr).$

Given that $\displaystyle \sum_{n=1}^\infty u_n = 30$ and the common ratio $r$ satisfies $\tfrac{1}{r} + 1 = 5$, find $u_1$.