Embibe Experts Solutions for Chapter: Logarithms and Exponentials, Exercise 1: IOQM - PRMO and RMO 2019 (25-08-2019)

Consider the sequence $1, 7, 8, 49, 50, 56, 57, 343, .........$ which consists of sums of distinct powers of $7$ that is $7^0, 7^1, 7^0+7^1, 7^2, .......$ in increasing order. At what position will $16856$ occurs in this sequence?

In base $-2$ notation, digits are $0$ and $1$ only and the places go up in powers of $-2$. For example, $11011$ stands for ${\left(-2\right)}^4+{\left(-2\right)}^3+{\left(-2\right)}^1+{\left(-2\right)}^0$ and equals number $7$ in base $10$. If the decimal number $2019$ is expressed in base $-2$, how many non-zero digits does it contain?