The 2nd and 5th terms of a GP are -7 and 56 respectively. Find:
a. The common ratio
b. The first term
c. The sum of the first five terms
a2 = - 7 , a5 = 56
nth term in GP:
an = a1 ∙ r ⁿ⁻¹
where:
a1 = first term , r = common ratio
a2 = a1 ∙ r ²⁻¹ = r¹ = a1 ∙ r
a5 = a1 ∙ r ⁵⁻¹ = a1 ∙ r ⁴
a5 = a1 ∙ r ∙ r ³
a5 = a2 ∙ r ³
56 = ( - 7 ) ∙ r ³
Divide both sides by - 7
56 / ( - 7 ) = r ³
- 8 = r ³
r ³ = - 8
r = ∛ - 8
r = - 2
a2 = a1 ∙ r
- 7 = a1 ∙ ( - 2 )
Divide both sides by - 2
- 7 / - 2 = a1 ∙ ( - 2 ) / - 2
7 / 2 = a1
a1 = 7 / 2
The sum of the n terms in GP:
Sn = a1 ∙ ( 1 - rⁿ ) / ( 1 - r )
The sum of the first five terms:
S5 = a1 ∙ ( 1 - r⁵ ) / ( 1 - r )
S5 = ( 7 / 2 ) ∙ [ 1 - ( - 2⁵ ) ] / [ 1 - ( - 2 ) ]
S5 = ( 7 / 2 ) ∙ [ 1 - ( - 32 ) ] / ( 1 + 2 )
S5 = ( 7 / 2 ) ∙ ( 1 + 32 ) / 3
S5 = ( 7 / 2 ) ∙ 33 / 3
S5 = ( 7 / 2 ) ∙ 11
S5 = 77 / 2 = 38.5
Your GP:
7 / 2 , - 7 , 14 , - 28 , 56...
Alphabet B
2nd term=-7
5th term=56
r=-2
Substitute -2 for r in (1)
a=-7
a=1.75
a. The common ratio:
To find the common ratio (r), we divide the 5th term by the 2nd term:
r = 56 / (-7) = -8.
b. The first term:
Let's call the first term of the geometric progression (GP) "a".
The 2nd term (T_2) is given as -7, and the common ratio (r) is -8, so we can use the formula for the nth term to find the first term (T_1):
T_2 = a * r^(2-1),
-7 = a * (-8)^1,
-7 = -8a,
a = (-7) / (-8),
a = 7/8.
c. The sum of the first five terms:
The formula to find the sum of the first n terms of a geometric progression is:
S_n = a * (1 - r^n) / (1 - r),
For the first five terms (n = 5), we substitute a = 7/8 and r = -8 into the formula:
S_5 = (7/8) * (1 - (-8)^5) / (1 - (-8)).
Now, simplify the expression:
S_5 = (7/8) * (1 - 32768) / 9,
S_5 = (7/8) * (-32767) / 9,
S_5 = -28889/8.
So, the sum of the first five terms is -28889/8.
To solve this problem, we need to use the formula for the nth term of a geometric progression (GP), which is given by:
\[a_n = a_1 \cdot r^{(n-1)}\]
Where:
- \(a_n\) is the nth term of the GP
- \(a_1\) is the first term of the GP
- \(r\) is the common ratio of the GP
- \(n\) is the position of the term in the GP
Let's solve each part of the problem step by step:
a. To find the common ratio (r), we can use the given information that the 2nd term is -7 and the 5th term is 56. We can set up two equations using the formula above:
Equation 1: \[-7 = a_1 \cdot r^{(2-1)}\]
Equation 2: \[56 = a_1 \cdot r^{(5-1)}\]
Simplifying Equation 1:
\[-7 = a_1 \cdot r\]
Simplifying Equation 2:
\[56 = a_1 \cdot r^4\]
Now, we can divide Equation 2 by Equation 1 to eliminate \(a_1\):
\[\frac{56}{-7} = \frac{a_1 \cdot r^4}{a_1 \cdot r}\]
\[-8 = r^3\]
Taking the cube root of both sides:
\[-2 = r\]
So, the common ratio (r) is -2.
b. To find the first term (a1), we can substitute the value of r (-2) into Equation 1:
\[-7 = a_1 \cdot (-2)\]
Simplifying:
\[a_1 = \frac{-7}{-2}\]
\[a_1 = 3.5\]
Therefore, the first term (a1) is 3.5.
c. To find the sum of the first five terms, we can use the formula for the sum of the first n terms of a GP, which is given by:
\[S_n = \frac{a_1 \cdot (1-r^n)}{1-r}\]
Substituting the values we found into this formula:
\[S_5 = \frac{3.5 \cdot (1-(-2)^5)}{1-(-2)}\]
Simplifying:
\[S_5 = \frac{3.5 \cdot (1-32)}{1+2}\]
\[S_5 = \frac{3.5 \cdot (-31)}{3}\]
\[S_5 = -35\]
Therefore, the sum of the first five terms is -35.