(a) The sum of the sixth terms of GP is 252
(b) The common difference and the first term of the AP are 3 and 4 respectively
Step-by-step explanation:
The formula of the nth term of an Geometric Progression (GP) is
[tex]a_{n}=ar^{n-1}[/tex] , where
- a is the first term
- r is the common ratio between each two consecutive terms
The formula of the sum of n terms of an Geometric Progression is
[tex]S_{n}=\frac{a(r^{n}-1)}{r-1}[/tex]
The formula of the nth term of an Arithmetic Progression (AP) is
[tex]a_{n}=a+(n-1)d[/tex] , where
- a is the first term
- d is the common difference between each two consecutive terms
The formula of the sum of n terms of an Arithmetic Progression is
[tex]S_{n}=\frac{n}{2}[2a+(n-1)d][/tex]
∵ A GP has the first term and common ratio 4 and 2 respectively
∴ a = 4 and r = 2
- Find the third term of it
∵ [tex]a_{n}=ar^{n-1}[/tex]
∵ n = 3
- Substitute the values of a, r, n in the formula to find the 3rd term
∴ [tex]a_{3}=4(2)^{3-1}=4(2)^{2}=4(4)=16[/tex]
∴ The third term is 16
∵ The sum of n terms in GP is [tex]S_{n}=\frac{a(r^{n}-1)}{r-1}[/tex]
∵ n = 6 , a = 4 and r = 2
- Substitute the values of a, r, n in the formula to find [tex]S_{6}[/tex]
∴ [tex]S_{6}=\frac{4[(2)^{6}-1]}{2-1}[/tex]
∴ [tex]S_{6}=\frac{4[64-1]}{1}[/tex]
∴ [tex]S_{6}=4[63][/tex]
∴ [tex]S_{6}=252[/tex]
(a) The sum of the sixth terms of GP is 252
∵ The third term of the GP is the fifth term of an AP
∵ The third term of the GP = 16
∴ The 5th term of AP = 16
∵ [tex]a_{n}=a+(n-1)d[/tex]
∵ n = 5
∵ [tex]a_{5}=a+(5-1)d[/tex]
∴ [tex]a_{5}=a+4d[/tex]
∵ [tex]a_{5}=16[/tex]
- Equate the two expressions of [tex]a_{5}[/tex]
∴ a + 4d = 16 ⇒ (1)
∵ The sum of the first ten terms of the AP is 175
∴ n = 10
∵ [tex]S_{10}=\frac{10}{2}[2a+(10-1)d][/tex]
∴ [tex]S_{10}=5[2a+9d][/tex]
∵ [tex]S_{10}=175[/tex]
- Equate the two expressions of [tex]S_{10}[/tex]
∴ 5[2a + 9d] = 175
- Divide both sides by 5 to simplify the equation
∴ 2a + 9d = 35 ⇒ (2)
Now we have a system of equation to solve it
Multiply equation (1) by -2 to eliminate a
∵ -2a - 8d = -32 ⇒ (3)
- Add equations (2) and (3)
∴ d = 3
- Substitute the value of d in equation (1) or (2) to find a
∵ a + 4(3) = 16
∴ a + 12 = 16
- Subtract 12 from both sides
∴ a = 4
(b) The common difference and the first term of the AP are 3 and 4 respectively
Learn more:
You can learn more about the AP and GP in brainly.com/question/7221312
brainly.com/question/1522572
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