Question

Help please will give you points

Answer 1

Answer:

16

Step-by-step explanation:

From the question given above, the following data were obtained:

Sum of N term (Sₙ) = 292

2nd term (T₂) = 8.5

5th term (T₅) = 13

Number of terms (N) =?

Next, we shall determine the first term (a) and the common difference (d) of the series. This can be obtained as follow:

T₂ = a + d

8.5 = a + d ........ 1

T₅ = a + 4d

13 = a + 4d ........ 2

Subtract equation 1 from equation 2 i.e

(2) – (1)

.. 13 = a + 4d

– 8.5 = a + d

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

4.5 = 3d

Divide both side by 3

d = 4.5 / 3

d = 1.5

Substitute the value of d into any of the equation to obtain the value of a. Here, we shall use equation 1.

8.5 = a + d

d = 1.5

8.5 = a + 1.5

Collect like terms

8.5 – 1.5 = a

a = 7

Finally, we shall determine the number of terms as follow:

Sum of N term (Sₙ) = 292

Common difference (d) = 1.5

First term (a) = 7

Number of terms (N) =?

Sₙ = n/2[2a + (n – 1)d]

292 × 2 = n[(2 × 7) + (n – 1)1.5]

584 = n[14 + 1.5n – 1.5]

584 = n[12.5 + 1.5n]

584 = 12.5n + 1.5n²

Rearrange

1.5n² + 12.5n – 584 = 0

Using formula method, the value of n can be obtained as follow:

a = 1.5

b = 12.5

c = –584

n = –b ± √(b² – 4ac) / 2a

n = –12.5 ± √(12.5² – 4 × 1.5 × –584) / 2 × 1.5

n = –12.5 ± √(156.25 + 3504) / 3

n = –12.5 ± √(3660.25) / 3

n = –12.5 ± 60.5 / 3

n = –12.5 + 60.5 / 3 or –12.5 – 60.5 / 3

n = 4 /3 or –73 / 3

n = 16 or –73 / 3

Since the number of terms can not be negative, therefore, the number of term is 16