The sum of first 13th term as G.P. Is 21 and the sum of 21th term in 13. find the sum of first 34th terms.
a(r^13 - 1)/(r-1) = 21 (#1)
a(r^21 - 1)/(r-1) = 13 (#2)
divide #2 by #1, the "a" will cancel, so will the (r-1) to get
(r^21 - 1)/(r^13 - 1) = 13/21
I then cross-multiplied and simplified to get
21r^21 - 13r^13 = 8
At this point I am currently at a stand-still.
Perhaps a different approach?
I noticed that 34 = 21 + 13
I'll try to show that the equation of Reiny 21r^21-13r^13=8 has only one real solution r=1.
Let F(r)=21r^21-13r^13-8, F(1)=0.
F'(r)=441r^20-169r^12=
=169r^12(441/169r^8-1)=169r^2(21/13r^4+1)*
(21/13r^4-1)=169r^2(21/13r^4+1)* (sqrt(21/13)r^2+1)(sqrt(21/13)r^2-1)
F'(r)=0 if r=r1=-(13/21)^0.25, r=r2=0, or r=r3=(13/25)^0.25
Fmax=F(r1)=-1.693443+2.735-8<0 Q.E.D.
I am surprised!
I looked at the situation where r = 1
It satisfies the equation 21r^21 - 13r^13 = 0
but in the original formula for the sum of a GS, that would make the denominator zero, thus undefined.
From the terms of problem => r=1 doesn't satisfy (S13=21, S21=13).
From my proof => such G.P. doesn't exist.
To find the sum of the first 13 terms of a geometric progression (G.P.), we can use the formula:
Sum of n terms of a G.P. = a * (r^n - 1) / (r - 1)
where:
- a is the first term of the G.P.
- r is the common ratio of the G.P.
- n is the number of terms in the series.
In this case, we have the sum of the 13th term of the G.P. equal to 21. So we can set up the equation as follows:
21 = a * (r^13 - 1) / (r - 1) .....(Equation 1)
Similarly, we know that the sum of the 21st term in the G.P. is 13. We can set up another equation:
13 = a * (r^21 - 1) / (r - 1) .....(Equation 2)
We now have two equations with two variables (a and r), we can solve them simultaneously to find their values.
Let's rearrange Equation 1 to solve for a:
a = (21 * (r - 1)) / (r^13 - 1)
Now substitute this value of a in Equation 2:
13 = [(21 * (r - 1)) / (r^13 - 1)] * [(r^21 - 1) / (r - 1)]
Now, simplify the equation by canceling out common terms and cross multiplying to get rid of the denominators. Then rearrange the equation in a form that allows us to find the value of r.
After solving for r, substitute its value back into either Equation 1 or 2 to find the value of a.
Finally, once you have found the values of a and r, you can use the formula to find the sum of the first 34 terms of the G.P.:
Sum of 34 terms = a * (r^34 - 1) / (r - 1)