Answer:
16
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