The fifth term of an eponential sequence is 4375 and the second term is 35. Find the third , sixth term and sum of the of the first five term
translate into math:
The fifth term of an eponential sequence is 4375 ----> ar^4 = 4375
the second term is 35 ----> ar = 35
divide the first equation by the second:
ar^4/(ar) = 4375/35
r^3 = 125
r = 5 <----- the cube root of r^3 is r, the cube root of 125 is 5
now sub that into ar=35 to find a
the rest is routine by simply using your formulas for "sixth term" and the "sum of first five terms?
an = a1 ∙ r ⁿ⁻¹
a2 = a1 ∙ r ²⁻¹
a2 = a1 ∙ r¹
a2 = a1 ∙ r = 35
a5 = a1 ∙ r ⁵⁻¹
a5 = a1 ∙ r⁴
4375 = a1 ∙ r ∙ r³
4375 = 35 ∙ r³
r³ = 4375 / 35 = 125
r = ∛125
r = 5
a1 ∙ r = 35
a1 ∙ 5 = 35
a1 = 35 / 5 = 7
a1 = 7
Now use formula:
an = a1 ∙ r ⁿ⁻¹
to find other terms
and formula for nth partial sum of a geometric sequence:
Sn = a1 ( 1 - rⁿ ) / ( 1 - r )
S5 = a1 ∙ ( 1 - 5⁵ ) / ( 1 - 5 )
S5 = 7 ∙ ( 1 - 3125 ) / - 4
S5 = 7 ∙ ( - 3124 ) / - 4
S5 = 5467
To solve this problem, we need to determine the common ratio (r) of the exponential sequence. Once we have the common ratio, we can find any term of the sequence using the formula nth term = a * r^(n - 1), where a is the first term and n is the position of the term.
Step 1: Find the common ratio (r).
We are given the second term (35) and the fifth term (4375).
The fifth term is the second term multiplied by the common ratio four times.
Therefore, 4375 = 35 * r^4.
To isolate the common ratio, we divide both sides of the equation by 35:
4375 / 35 = r^4.
Simplifying the left side:
125 = r^4.
Take the fourth root of both sides to solve for r:
∜125 = r.
The fourth root of 125 is 5, so the common ratio (r) is 5.
Step 2: Find the third term.
The third term can be found using the formula nth term = a * r^(n - 1).
Substituting the values:
Third term = 35 * 5^(3 - 1)
Third term = 35 * 5^2
Third term = 35 * 25
Third term = 875.
Therefore, the third term is 875.
Step 3: Find the sixth term.
The sixth term can be found using the formula nth term = a * r^(n - 1).
Substituting the values:
Sixth term = 35 * 5^(6 - 1)
Sixth term = 35 * 5^5
Sixth term = 35 * 3125
Sixth term = 109375.
Therefore, the sixth term is 109375.
Step 4: Find the sum of the first five terms.
To find the sum of the first five terms, we can use the formula for the sum of an exponential sequence: sum = a * (1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms.
Substituting the values:
Sum = 35 * (1 - 5^5) / (1 - 5)
Sum = 35 * (1 - 3125) / (1 - 5)
Sum = 35 * (-3124) / (-4)
Sum = 35 * 781
Sum = 27335.
Therefore, the sum of the first five terms is 27335.
To summarize:
- The value of the third term is 875.
- The value of the sixth term is 109375.
- The sum of the first five terms is 27335.