Question

Σ(6 – 4n) = -48
Find the value of n:

A) 12
B) 14
C) 16
D) 18

Answer 1

The value of n is found by rearranging the summation equation Σ(6 – 4n) = -48 to solve for the number of terms, resulting in 12 when substituted back into the equation, aligning with option A.

Step-by-step explanation:

The summation Σ(6 – 4n) = -48 represents the sum of the expression 6 - 4n for a range of values of n. To solve for n, set the sum equal to -48 and solve for n. Reorganizing the equation gives Σ(6) - Σ(4n) = -48. Now, Σ(6) represents a constant multiplied by the number of terms in the sequence, while Σ(4n) is a sum involving n. Since Σ(6) equals 6 times the number of terms, and -48 is the total sum, the number of terms times 6 minus the sum of 4n equals -48. Solving this gives the number of terms as 10, and when substituted back into the equation, it leads to n = 12, which aligns with option A.

The equation Σ(6 – 4n) = -48 can be rearranged to Σ(6) - Σ(4n) = -48. Here, Σ(6) represents the sum of 6 across a range of terms, and Σ(4n) denotes the sum involving 4n across the same range. The difference between Σ(6) and Σ(4n) results in -48. Knowing that Σ(6) equals 6 multiplied by the number of terms, we can express this equation as 6n - Σ(4n) = -48, where n represents the number of terms.

By isolating Σ(4n), which is equivalent to 4 times the sum of the values of n across the range, we get 6n + 48 = Σ(4n). Since Σ(4n) equals 4 times the sum of n across the range, and it's also equal to 6n + 48, we can solve for n. Setting 6n + 48 = -48, we find n = 12, which is in line with option A, making 12 the value of n.

Answer 2

C) 16

The expression Σ(6 – 4n) simplifies to Σ6 - Σ4n, resulting in Σ4n = -54. Dividing by 4 yields Σn = -13.5, and the closest integer solution is n = 16, making the summation equal to -48.

Step-by-step explanation:

The given equation is Σ(6 – 4n) = -48. To find the value of n, we can simplify the expression inside the summation. The expression 6 – 4n represents a sequence of numbers obtained by substituting different values of n. By expanding the summation, we get Σ6 - Σ4n = -48. Simplifying further, the sum of 6 (which is 6 * Σ1) subtracted from the sum of 4n leads to Σ4n = -54. Dividing both sides by 4, we find that Σn = -54/4 = -13.5.

However, the value of n must be an integer, so we need to look for the closest integer value. Since n is multiplied by 4, we want the sum to be close to, but not exceeding, -13.5. The only integer that satisfies this condition is n = 16. Therefore, the correct answer is C) 16.

In summary, the solution involves simplifying the expression, finding the sum of the series, and then determining the closest integer value for n that meets the given condition. The final answer is n = 16, which makes the summation equal to -48.