Question

If seven times the seventh term of an AP is equal to eleven times its eleventh term, its 18 th term is (a) \(-23\) (b) \(-15\) (c) 16 (d) 0

Short answer

Answer: The 18th term of the progression is equal to the negative of the first term (-a1). We cannot determine the exact value of the 18th term without knowing the first term.

Step-by-step solution

Write the formula for the n-th term of an AP

The general formula for the n-th term of an AP can be expressed as: $$ a_n = a_1 + (n-1)d $$ where \(a_n\) is the n-th term, \(a_1\) is the first term, \(n\) is the number of terms, and \(d\) is the common difference between the terms.

Use the given condition to form an equation

We are given that seven times the seventh term is equal to eleven times the eleventh term. We can write this condition as: $$ 7a_7=11a_{11} $$ Now, we can replace \(a_7\) and \(a_{11}\) with their respective formulas from Step 1: $$ 7(a_1 + 6d)= 11(a_1 + 10d) $$

Solve the equation for the common difference d

Now, we will solve the equation from Step 2 for \(d\): $$ 7a_1 + 42d = 11a_1 + 110d $$ Rearrange the equation to isolate the variable \(d\): $$ 110d - 42d = 7a_1 - 11a_1 $$ Combine the like terms: $$ 68d = -4a_1 $$ Now, divide both sides by 68: $$ d = \frac{-4a_1}{68} $$

Substitute the expression for d into the formula for the 18th term

Now that we have the expression for \(d\), we can find the 18th term of the AP. We will use the formula for the n-th term and substitute the expression for \(d\): $$ a_{18} = a_1 + 17d $$ Substituting the expression for \(d\) into the formula: $$ a_{18} = a_1 + 17\left(\frac{-4a_1}{68}\right) $$ Simplify the equation: $$ a_{18} = a_1 - \frac{68a_1}{68} $$ Now, cancel out the common factors: $$ a_{18} = -a_1 $$

Conclusion

From the above analysis, we find that the 18th term is equal to the negative of the first term in the AP. Since we don't have the values of \(a_1\), we cannot determine the exact value of the 18th term. Therefore, none of the given options are correct, as we need more information to determine the exact value of the 18th term.

Key concepts

nth-term formula

In arithmetic progression (AP), the nth-term formula is a powerful tool. It helps us find any term in the sequence without listing every preceding term. The formula is \( a_n = a_1 + (n-1)d \). Here, \(a_n\) represents the term we want, \(a_1\) is the first term of the sequence, \(n\) is the term number, and \(d\) is the common difference between consecutive terms.

Using this formula, you can quickly calculate any term's value by knowing just the first term and the common difference. This eliminates the need to add up each term sequentially, saving time and effort.

common difference

The common difference in an arithmetic progression is the amount that each term differs from the previous one. It is constant throughout the sequence and denoted by \(d\). This value is crucial as it defines the pattern of the progression.

For instance, if you have terms 3, 6, 9, the common difference \(d\) is 3, since \(6 - 3 = 9 - 6 = 3\). Recognizing the common difference helps in predicting all other values in the sequence, starting from any known term.

• It adds uniformity, ensuring that the sequence grows or declines at a constant rate.
• Understanding \(d\) allows for the use of the nth-term formula effectively by providing necessary information about the pattern of growth or decrease.

sequence equation

A sequence equation is formed when we use known terms and conditions to create an algebraic representation of the sequence. In the context of arithmetic progression, this equation helps us find unknown values such as the common difference or specific terms in the sequence.

For example, if we know that seven times the seventh term equals eleven times the eleventh term, we formulate the sequence equation: \( 7(a_1 + 6d) = 11(a_1 + 10d) \). This comes directly from applying the nth-term formula for both the 7th and 11th terms in the sequence.

Solving this equation involves utilizing properties of equality and algebraic manipulation to isolate and find the required unknowns, like \(d\). It simplifies finding relationships between different terms of the sequence.

term substitution

Term substitution is a method where we replace known values or expressions into a formula to get desired outcomes. In arithmetic progression, it allows us to use the relationships between terms to find unknown values effortlessly.

For instance, after solving for the common difference \(d\) from a sequence equation, substituting \(d\) back into the nth-term formula allows us to find specific sequence terms like \(a_{18}\).

In the given exercise solution, after determining \(d\), we substitute it into the formula for the 18th term: \( a_{18} = a_1 + 17d \). This substitution helps us express the 18th term in terms of the first term, leading to a deeper understanding of the sequence's structure even when certain values remain unknown.