Understanding the Arithmetic Series Problem

The question asks for the number of terms in a specific number series. The series is given as 7, 14, 21, 28, and so on. We are also given that the sum of this series up to a certain number of terms is 952.

First, let's identify the type of series we are dealing with. Observing the terms:

• The first term is 7.
• The difference between the second term (14) and the first term (7) is $14 - 7 = 7$.
• The difference between the third term (21) and the second term (14) is $21 - 14 = 7$.
• The difference between the fourth term (28) and the third term (21) is $28 - 21 = 7$.

Since the difference between consecutive terms is constant (which is 7), this series is an Arithmetic Progression (AP).

Key Information Identified

• First term (a) = 7
• Common difference (d) = 7
• Sum of the series ($S_n$) = 952
• We need to find the number of terms (n).

Using the Sum Formula for Arithmetic Progression

The formula for the sum of the first 'n' terms of an arithmetic progression is:

$$ S_n = \frac{n}{2} [2a + (n-1)d] $$

We have $S_n = 952$, $a = 7$, and $d = 7$. Let's substitute these values into the formula:

$$ 952 = \frac{n}{2} [2(7) + (n-1)7] $$

Now, we simplify the equation to solve for 'n':

$$ 952 = \frac{n}{2} [14 + 7n - 7] $$

$$ 952 = \frac{n}{2} [7 + 7n] $$

We can take 7 common from the terms inside the bracket:

$$ 952 = \frac{n}{2} [7(1 + n)] $$

$$ 952 = \frac{7n(1 + n)}{2} $$

Multiply both sides by 2:

$$ 952 \times 2 = 7n(1 + n) $$

$$ 1904 = 7n + 7n^2 $$

Divide the entire equation by 7:

$$ \frac{1904}{7} = \frac{7n + 7n^2}{7} $$

$$ 272 = n + n^2 $$

Rearrange the equation into a standard quadratic form ($an^2 + bn + c = 0$):

$$ n^2 + n - 272 = 0 $$

Solving the Quadratic Equation for the Number of Terms

We need to find the value of 'n' that satisfies this quadratic equation. We can factor the quadratic equation or use the quadratic formula. Alternatively, since we have options for 'n', we can test each option.

Method 1: Testing the Options

Let's test the given options for 'n' using the sum formula or the derived equation $n^2 + n - 272 = 0$. The options are 16, 17, 18, and 19.

• Testing n = 16 (Option A): Substitute n=16 into $n^2 + n - 272 = 0$ $16^2 + 16 - 272 = 256 + 16 - 272 = 272 - 272 = 0$. Since the equation holds true, n=16 is a valid solution. Let's verify using the sum formula: $S_{16} = \frac{16}{2} [2(7) + (16-1)7] = 8 [14 + 15 \times 7] = 8 [14 + 105] = 8 [119] = 952$. This matches the given sum.
• Testing n = 17 (Option B): Substitute n=17 into $n^2 + n - 272 = 0$ $17^2 + 17 - 272 = 289 + 17 - 272 = 306 - 272 = 34$. Since the result is not 0, n=17 is not the correct number of terms for the sum 952.
• Testing n = 18 (Option C): Substitute n=18 into $n^2 + n - 272 = 0$ $18^2 + 18 - 272 = 324 + 18 - 272 = 342 - 272 = 70$. Since the result is not 0, n=18 is not the correct number of terms for the sum 952.
• Testing n = 19 (Option D): Substitute n=19 into $n^2 + n - 272 = 0$ $19^2 + 19 - 272 = 361 + 19 - 272 = 380 - 272 = 108$. Since the result is not 0, n=19 is not the correct number of terms for the sum 952.

Method 2: Factoring the Quadratic Equation

We need to find two numbers that multiply to -272 and add up to 1 (the coefficient of n). After trying factors of 272 (like 1, 2, 4, 8, 16, 17, 34, etc.), we find that 17 and -16 satisfy the conditions (17 * -16 = -272 and 17 + (-16) = 1).

So, the equation $n^2 + n - 272 = 0$ can be factored as:

$$ (n + 17)(n - 16) = 0 $$

This gives two possible values for n:

• $n + 17 = 0 \implies n = -17$
• $n - 16 = 0 \implies n = 16$

Since the number of terms in a series cannot be negative, we discard $n = -17$. Therefore, the valid number of terms is $n = 16$.

Conclusion on the Number of Terms

Both methods lead to the conclusion that the number of terms in the series whose sum is 952 is 16.

Comparing this result with the given options, option A is 16.

Revision Table: Key Arithmetic Progression Concepts

Additional Information on Arithmetic Series and Sum

An arithmetic series is a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the common difference, denoted by 'd'. The first term is denoted by 'a'.

The series in this problem, 7, 14, 21, 28, ... is an arithmetic series where $a=7$ and $d=7$. Each term is simply 7 times the term number (e.g., the 3rd term is $7 \times 3 = 21$). The n-th term ($a_n$) can also be found using the formula $a_n = a + (n-1)d$. For this specific series, $a_n = 7 + (n-1)7 = 7 + 7n - 7 = 7n$. So the series is simply the multiples of 7.

The sum of an arithmetic series can be a useful concept in various mathematical and real-world problems, such as calculating total earnings with a fixed increase each period or finding the sum of consecutive numbers.

When solving problems involving the sum of an arithmetic series and needing to find the number of terms, it often leads to a quadratic equation in terms of 'n'. Remember that 'n' must always be a positive integer.