the 5 term of a gp is 4375 and the 2 term is 35 find (a)the 3 (b)the 6 (c)the sum of the first five terms
ar^4 = 4375
ar = 35
divide, and you get
r^3 = 125
r = 5
Now you know that
a = 35/5 = 7
and S5 = a(r^5-1)/(r-1)
how
To find the missing terms and the sum of the first five terms in a geometric progression (GP) given two terms, we can use the formula for the nth term of a GP and the formula for the sum of the first n terms of a GP.
The formula for the nth term (Tn) of a GP is given by: Tn = a * r^(n - 1)
Where:
- a is the first term
- r is the common ratio
- n is the term number
Let's solve the given problem step by step:
Given:
- T2 = 35
- T5 = 4375
Step 1: Find the common ratio (r)
Since we have T2 and T5, we can work backward to find the common ratio.
From the formula, we have:
T2 = a * r^(2 - 1) = a * r
T5 = a * r^(5 - 1) = a * r^4
Dividing T5 by T2:
(T5 / T2) = (a * r^4) / (a * r) = r^3
Substituting the given values:
35 / 4375 = r^3
Simplifying the equation:
1 / 125 = r^3
Taking the cube root of both sides:
∛(1 / 125) = ∛(r^3)
1 / 5 = r
So, the common ratio (r) is 1/5.
Step 2: Find the first term (a)
Since we have the common ratio (r) and T2, we can find the first term.
From the formula, we have:
T2 = a * r^(2 - 1) = a * r
35 = a * (1/5)
Multiplying both sides by 5:
5 * 35 = a
175 = a
So, the first term (a) is 175.
Step 3: Find the third term (T3)
Using the formula for the nth term:
T3 = a * r^(3 - 1)
T3 = 175 * (1/5)^2
T3 = 175 * (1/25)
T3 = 7
So, the third term (T3) of the GP is 7.
Step 4: Find the sixth term (T6)
Using the formula for the nth term:
T6 = a * r^(6 - 1)
T6 = 175 * (1/5)^5
T6 = 175 * (1/3125)
T6 = 0.056
So, the sixth term (T6) of the GP is approximately 0.056.
Step 5: Find the sum of the first five terms (S5)
The sum of the first n terms of a GP is given by the formula:
S_n = a * (1 - r^n) / (1 - r)
From the given information, we have:
a = 175 (first term)
r = 1/5 (common ratio)
n = 5 (number of terms)
Using the formula:
S5 = 175 * (1 - (1/5)^5) / (1 - 1/5)
Simplifying the equation:
S5 = 175 * (1 - 1/3125) / (4/5)
S5 = 175 * (3125 - 1) / 3125 * (5/4)
S5 = 175 * 3124 / 3125 * (5/4)
S5 = 2187500 / 12500
S5 = 175
So, the sum of the first five terms (S5) is 175.
Therefore, the answers to the given questions are:
(a) The third term (T3) is 7.
(b) The sixth term (T6) is approximately 0.056.
(c) The sum of the first five terms (S5) is 175.