Find the sum of the squares of the terms of an infinite geometric sequence given the first term is 49 and the common ratio is nine 𝑥.
Let’s begin by recalling what we know about a geometric sequence. The first term of a geometric sequence is denoted by the letter 𝑎. The common ratio of a geometric sequence is equal to 𝑟. This is the number that we multiply by to get from the first to the second term, the second to the third term, and so on. This means that any geometric sequence has the terms 𝑎, 𝑎𝑟, 𝑎𝑟 squared, and so on. The 𝑛th term of a geometric sequence is equal to 𝑎 multiplied by 𝑟 to the power of 𝑛 minus one.
In this question, we’re asked to find the sum of an infinite geometric sequence. The sum to ∞ of any geometric sequence is calculated by using the formula 𝑎 divided by one minus 𝑟. This is only true or valid, however, if the absolute value or modulus of 𝑟 is less than one. We’re told in the question that our sequence has a first term 𝑎 equal to 49. It has a common ratio equal to nine 𝑥.
Our first thought might be to substitute these values into our sum to ∞ formula. The sum to ∞ is equal to 49 divided by one minus nine 𝑥. This question asks for the sum of the squares of the terms in the sequence. We might therefore be tempted to square the sum to ∞ expression. Squaring the numerator would give us 2401. And squaring the denominator would give us one minus nine 𝑥 all squared. This is not the correct method for this question though.
We need to find the actual sequence of the squares and then find the sum to ∞ of this. This is because the squares of the geometric sequence are themselves a geometric sequence. The sum to ∞ formula is therefore valid. We need to calculate 𝑎 squared and 𝑟 squared. 𝑎 squared is equal to 49 squared. As already stated, this is equal to 2401. 𝑟 squared is equal to nine 𝑥 squared. Nine squared is equal to 81. Therefore, 𝑟 squared is equal to 81𝑥 squared. As this is also a geometric sequence, we can substitute these values into the sum to ∞ formula.
The sum to ∞ is therefore equal to 2401 divided by one minus 81𝑥 squared. This is different to finding the sum to ∞ of the original sequence and then squaring it.