If&nbsp;${a_{1},a_{2},a_{3},\ldots}$ are terms of a geometric sequence with common ratio ${r\neq 1}$, show that
${(a_{1}-a_{2})^{2}+(a_{2}-a_{3})^{2}+(a_{3}-a_{4})^{2}+\cdots+(a_{n}-a_{n+1})^{2}=\frac{a_{1}^{2}(1-r)(1-r^{2n})}{1+r}}$.[7]

If ${a_{1},a_{2},a_{3},\ldots}$ are terms of a geometric sequence with common ratio ${r\neq 1}$, show that

${(a_{1}-a_{2})^{2}+(a_{2}-a_{3})^{2}+(a_{3}-a_{4})^{2}+\cdots+(a_{n}-a_{n+1})^{2}=\frac{a_{1}^{2}(1-r)(1-r^{2n})}{1+r}}$.

Sequence and Series - Geometric Progression

Sequence and Series - Geometric Progression

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