In each of the following, the nth term of a sequence is given. Find the first 4 terms, a10, and

In each of the following, the nth term of a sequence is...

Key Concepts

Sequence

A sequence is an ordered list of numbers where each number is called a term. In this context, the sequence is defined by a formula that generates its nth term, which allows us to understand the overall pattern or rule governing the sequence.

Nth Term Expression

The nth term expression is a formula that directly relates the position of a term in the sequence to its value. This allows one to calculate any term in the sequence without needing to compute the previous terms, making it a powerful tool in sequence analysis.

Substitution

Substitution is the process of replacing the variable 'n' with a specific number in the nth term expression to compute the corresponding term. This basic technique in sequences is essential for evaluating the numerical value of specific terms in the sequence.

Properties of Exponents

Understanding the properties of exponents is crucial when dealing with expressions that include exponential terms. This involves rules about how powers work, especially when bases are raised to different exponents, and helps simplify the expression before or after substitution.

Alternating Signs

An expression with a negative base raised to a variable exponent can result in alternating signs depending on the exponent's parity. Recognizing this behavior is important when analyzing sequences, as it affects the sign and therefore the value of each term in the sequence.

Transcript

00:01 According to the question we have to find out the first 4 term and a10 and 15 we have given a and term is 5 plus minus 2 raise to power n plus 1 upon 2 raise to power n.

00:20 We know that when we find first 4 term we put the value of n is 1, 2, 3 and 4.

00:28 When we put the value of n is 1, there is a 1.

00:33 In that case, 5 plus minus 2 raised to power 1 plus 1 upon 2 raised to power 1.

00:41 On solving, there is 5.

00:44 Minus 2 raised to power 2 is 4 upon 2.

00:49 When we solve, there is 2.

00:51 The answer will be 7.

00:54 Similarly, we find out a 2.

00:56 In that case, we put the value of n is 2.

00:59 Minus 2 to power 2 plus 1 upon 2 raised to power 2 on solving we know that there is a 2 to power 3 in that case minus term when minus term power is a odd number the result will be a negative that's why we write 5 minus 8 by 4 on solving the answer will be 3 similarly, we find out a 3, in that case 5 plus minus 2, raise to power 3 plus 1 upon 2 raise to power 3.

01:40 On solving, there is 5.

01:42 In that case, the power is even.

01:45 The answer will be positive.

01:48 There is 16 by 8 on solving there is 7.

01:53 Similarly, we find out the value of a 4.

01:57 In that case, we put the value of n is 4 plus minus 2, rest to power 4 plus 1 upon 2 raise to power 4.

02:11 On solving, we know that there is 5 plus in that case minus 2 power is an odd number.

02:18 The answer will be a negative.

02:20 2 raise to power 5 is a 32 upon 2 raise to power 4 is a 16.

02:26 When we cut the answer will be.

02:31 Similarly, we find out a 10.

02:34 In that case, we put value of n is 10...