Question 1127671

the general form of a geometric progression(GP) is

{{{a(n) = ar^(n-1)}}} with {{{a}}} = first term, {{{r }}}is the common ratio

we have {{{a = x + 1}}}, then

{{{a[1] = x + 1}}}
{{{a[2] = x + 3 = (x + 1)*r}}}
{{{a[3] = x + 8 = (x + 1)r^2}}}

solve {{{a[2]}}} for {{{r}}}

{{{r = (x + 3) / (x + 1)}}}

solve {{{a[3]}}} for {{{r^2}}}

{{{r^2 = (x +8 ) / (x + 1)}}}

then, 

i) The Value of {{{x}}}

{{{(x +8) / (x + 1) = (x+ 3)^2 / (x + 1)^2}}}

cross multiply the fractions


{{{(x +8) * (x + 1)^2 = (x + 3)^2 * (x +1)}}}


{{{(x +8) * (x + 1) = (x + 3)^2 }}}


{{{ x^2 + x+8x+8 = x^2 + 6x+9 }}}

{{{  9x+8 =  6x+9 }}}

{{{  9x-6x=  9-8 }}}

{{{  3x=  1 }}}

{{{  x=  1/3 }}}

then

ii) The common ratio

{{{r = (1/3 + 3) / (1/3 + 1) = (10/3) / (4/3) = 10/4=5/2}}}


iii) The sum of the first 15 terms.

{{{a[1] = x+1 = 1/3 + 1 = 4/3}}}

the general form of a geometric progression(GP) is

{{{a[n] = (4/3)(5/2)^(n-1)}}}


the first 15 terms

{{{a[1] = 4/3}}}

{{{a[2] =(4/3)(5/2)=10/3}}}
{{{a[3] = (4/3)(5/2)^2=(4/3)(25/4)=25/3}}}
{{{a[4] = (4/3)(5/2)^3= (4/3)(125/8)=125/6}}}
{{{a[5] = (4/3)(5/2)^4= (4/3)(625/16)=625/12}}}
{{{a[6] = (4/3)(5/2)^5= (4/3)(3125/32)=3125/24}}}
{{{a[7] = (4/3)(5/2)^6= (4/3)(15625/64)=15625/48}}}
{{{a[8] = (4/3)(5/2)^7= (4/3)(78125/128)=78125/96}}}
{{{a[9] = (4/3)(5/2)^8= (4/3)(390625/256)=390625/192}}}
{{{a[10] = (4/3)(5/2)^9= (4/3)(1953125/512)=1953125/384}}}
{{{a[11] = (4/3)(5/2)^10= (4/3)(9765625/1024)=9765625/768}}}
{{{a[12] = (4/3)(5/2)^11= (4/3)(48828125/2048)=48828125/1536}}}
{{{a[13] = (4/3)(5/2)^12= (4/3)(244140625/4096)=244140625/3072}}}
{{{a[14] = (4/3)(5/2)^13= (4/3)(1220703125/8192)=1220703125/6144}}}
{{{a[15] = (4/3)(5/2)^14= (4/3)(6103515625/16384)=6103515625/12288}}}


the sum is:
{{{ 4/3+10/3+25/3+125/6+625/12+3125/24+15625/48+78125/96+390625/192+1953125/384+9765625/768+48828125/1536+244140625/3072+1220703125/6144+6103515625/12288=3390838373/4096}}} ≈ {{{827841.399}}}