Problem 30 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

The first and last terms of an AP are 1 and 11 . If the sum of its terms is 36 , then find the number of terms.

Step-by-Step Solution

Answer

The number of terms in the arithmetic progression is 6.

Step 1: Substitute values into the nth term formula#

We have the formula: \(l = a + (n-1)d\). We are given the first term (a) and the last term (l). We can substitute the values into this formula: \(11 = 1 + (n - 1)d\)

Step 2: Rearrange the formula to solve for d#

We want to get the common difference (d) on one side of the equation. To do this, rearrange the formula: \(d = \frac{11 - 1}{n - 1} = \frac{10}{n - 1}\)

Step 3: Substitute values into the sum of terms formula#

Now that we have found the expression for d, we can use the arithmetic progression sum formula to find the number of terms. We have the formula: \(S = \frac{n}{2}(a + l)\) We are given the sum of terms (S) and the first and last terms (a and l), so we can substitute the values into this formula: \(36 = \frac{n}{2}(1 + 11)\)

Step 4: Solve for the number of terms#

Now, we can solve for the number of terms (n) in the arithmetic progression. Simplify the equation: \(36 = \frac{n}{2}(12)\) \(36 = 6n\) Divide by 6 to find the number of terms (n): \(n = \frac{36}{6}\) \(n = 6\)

Step 5: Write the final answer#

The number of terms in the arithmetic progression is 6.

Key Concepts

nth term formula

In an arithmetic progression (AP), finding any specific term involves using the nth term formula. This formula is crucial in determining the exact value of a term at a particular position in the sequence. The nth term is calculated using the formula: \[ l = a + (n-1) \cdot d \]where:

• \( l \) is the nth term or last term.
• \( a \) is the first term of the sequence.
• \( n \) denotes the number of the term position.
• \( d \) represents the common difference between consecutive terms.

This formula gives us the relation between the first and any nth term in the arithmetic progression. By knowing any three values, you can calculate the fourth one.
For instance, if you know the first term \( a \), the common difference \( d \), and the position \( n \), you can readily find the nth term \( l \). This is especially useful for long sequences where listing each term is not feasible.

sum of terms formula

In an arithmetic progression, calculating the sum of a certain number of terms is often required. The sum of the first \( n \) terms is determined using the formula:\[ S = \frac{n}{2} \cdot (a + l) \]where:

• \( S \) is the sum of the first \( n \) terms.
• \( n \) is the number of terms.
• \( a \) is the first term of the sequence.
• \( l \) is the last term of the sequence.

This formula means you find the average of the first and last term, and multiply by the number of terms.
The simplicity of this formula reduces the complexity of computing large sums manually.
For example, if given the sum \( S \), first term \( a \), and the last term \( l \), you can determine the total number of terms \( n \).

common difference

The common difference in an arithmetic progression is the backbone of the sequence, dictating how the series will grow or shrink. The common difference is simply the amount that you add (or subtract) to move from one term to the next. It is represented by \( d \) and is calculated using:\[ d = \frac{l - a}{n - 1} \]where:

• \( d \) is the common difference.
• \( l \) is the last term.
• \( a \) is the first term.
• \( n \) is the number of terms.

The common difference tells us how smoothly or steeply an arithmetic progression climbs or descends. You can determine the common difference if you have the first term, the last term, and the total number of terms.
This allows for the construction or deconstruction of the sequence, allowing you to generate any number of terms from known values.