Geometric progression?

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a r^4 + a r^6 = 10 ................... equation (1)

a r = 16 a r ^5

1 = 16 r^4

r^4 = 1 / 16

common rate (ratio), r = ± ½

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plug in the computed value of r into eq (1);

a r^4 + a r^6 = 10

..................... .................... let's just use +½ because r is raised to even numbers

a ( ½ )^4 + a ( ½ )^6 = 10

a ( 1 / 16 ) + a ( 1 / 64 ) = 10

4a + a = 640

5a = 640

first term, a = 128

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for r = + ½ :

sum of the first 4 terms is :

n

Σ a r^k = a ( 1 - r^(n+1)) / ( 1 - r )

k=0

3

Σ 128 ( ½ )^k = 128 ( 1 - ( ½ )^(3+1)) / ( 1 - ½ )

k=0

...................... = 128 ( 1 - ( ½ )^4 ) / ½

...................... = 128 ( 1 - ( 1 / 16 ) / ½

...................... = 128 ( 15 / 16 ) / ½

...................... = 8 ( 15 ) / ½

...................... = 120 / ½

...................... = 240

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for r = – ½ :

sum of the first 4 terms is :

n

Σ a r^k = a ( 1 - r^(n+1)) / ( 1 - r )

k=0

3

Σ 128 ( –½ )^k = 128 ( 1 - ( –½ )^(3+1)) / ( 1 - (–½ ) )

k=0

...................... = 128 ( 1 - ( –½ )^4 ) / 1½

...................... = 128 ( 1 - ( 1 / 16 ) / 1½

...................... = 128 ( 15 / 16 ) / 1½

...................... = 8 ( 15 ) / 1½

...................... = 120 / 1½

...................... = 80

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ar=16ar^5 , 1/16=r^4 , r=+-1/2 , ar^4+ar^6=10 , a=128 , terms 128 , 64 , 32 , 16 sum=240 or terms 128 , -64 , 32 , -16 sum=80

i 1/2

ii 128

iii 128+64+32+16

1.) Common rate = (16)^(1/(6-2) = 4th root of 16 = 2

I cant help you with the rest I am afraid.