Question

How many terms of the sequence -12,-9,-6,-3, must be taken to make the sum 54

Answer 1

Answer:

The required number of terms is 12.

Step-by-step explanation:

Given sequence : -12, -9, -6, -3, ...

⇒ Arithmetic Progression is a sequence of terms where each term is obtained by adding a constant to the preceding term.

⇒ The given sequence is an Arithmetic Progression since each term is obtained by adding '3' to the preceding term.

first term, a = -12

common difference, d = 3

Sum of n terms of an A.P is given by,

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

We have to find the number of terms to be taken to make the sum 54.

$\sf 54=\dfrac{n}{2}[2(-12)+(n-1)(3)] \\\\ \sf 54 \times 2=n[-24+3n-3] \\\\ \sf 108=n[3n-27] \\\\ \sf 108=3n^2-27n \\\\ \sf 108=3(n^2-9n) \\\\ \sf 108/3 = n^2-9n \\\\ \sf n^2-9n-36=0$

Solving the quadratic equation,

n² - 9n - 36 = 0

n² - 12n + 3n - 36 = 0

n(n - 12) + 3(n - 12) = 0

(n - 12) (n + 3) = 0

• n = +12
• n = -3

n can't be negative, hence n = 12

∴ The required number of terms is 12

Answer 2

Step-by-step explanation:

The required number of terms is 12.

Step-by-step explanation:

Given sequence : -12, -9, -6, -3, ...

⇒ Arithmetic Progression is a sequence of terms where each term is obtained by adding a constant to the preceding term.

⇒ The given sequence is an Arithmetic Progression since each term is obtained by adding '3' to the preceding term.

first term, a = -12

common difference, d = 3

Sum of n terms of an A.P is given by,

\longmapsto \sf S_n=\dfrac{n}{2}[2a+(n-1)d]⟼S

n

=

2

n

[2a+(n−1)d]

We have to find the number of terms to be taken to make the sum 54.

\begin{gathered}\sf 54=\dfrac{n}{2}[2(-12)+(n-1)(3)] \\\\ \sf 54 \times 2=n[-24+3n-3] \\\\ \sf 108=n[3n-27] \\\\ \sf 108=3n^2-27n \\\\ \sf 108=3(n^2-9n) \\\\ \sf 108/3 = n^2-9n \\\\ \sf n^2-9n-36=0\end{gathered}

54=

2

n

[2(−12)+(n−1)(3)]

54×2=n[−24+3n−3]

108=n[3n−27]

108=3n

2

−27n

108=3(n

2

−9n)

108/3=n

2

−9n

n

2

−9n−36=0

Solving the quadratic equation,

n² - 9n - 36 = 0

n² - 12n + 3n - 36 = 0

n(n - 12) + 3(n - 12) = 0

(n - 12) (n + 3) = 0

n = +12

n = -3

n can't be negative, hence n = 12

∴ The required number of terms is 12