00:01
Hi there, in this question, there's a geometric progression and geometric sequence.
00:07
There are six terms, so the first term is 192.
00:12
Let's say, a .n is the geometric sequence.
00:20
And the first term is, which is, let's say this is a1, which is 192.
00:25
So the common ratio, let's say the r is equal to 1 .3, 1 .5, that means this is 3 over 2.
00:34
So there is an arithmetic progression.
00:37
Let's say this is bofn.
00:39
So we just define this bofn as arithmetic sequence.
00:48
So there are 21 terms.
00:52
Exactly there are six terms here.
00:55
So six terms.
00:57
And there are 21 terms.
01:01
And give the sum of all the terms in geometry.
01:05
So there are 21 terms.
01:06
Terms here and the sum of these terms is equal to they are equal to each other so first of all let's find the sum of the terms for the geometric sequence let's say s n is the sum of the geometric sequence and we have the formula which is a1 times one minus r to the power n over one minus r so the first term is 192 and the common ratio is 3 over 2 which means this is 1 minus 3 over 2 to the 6 because there are 6 terms and 1 minus 3 over 2 let's do the calculation here which is 192 and this is negative 1 over 2 let me just write in this way 3 to the power 6 over 64 minus 1 divided by this is 3 over 2 and minus 1 so that should be 192 times 3 to the power 6 minus 64 divided by which is 64 n times this is 1 over 2 let's multiply these and this is 32 so if i just simplify this which is equal to 6 so we got 6 times 3 to the power 6 minus 64 so this is the sum of the geometric sequence here what about the arithmetic sequence so for the sum of the arithmetic sequence we have the formula which is n over 2 times this is a 1 to a 1 plus n minus 1 times the common difference, which is d.
02:43
So right now we have the common difference, which is given as a 1 .5.
02:49
That means this is 3 over 2.
02:51
So the number of terms is 21.
02:54
So plug in 21 over 2 times.
02:56
This is 2a1, so we don't know the first term.
02:59
And plus 21 minus 1, which is 20 times 3 over 2.
03:03
That should be equal to 6 times 3 to the power 6 minus 64, because there are some, should be equal to each other.
03:12
So let's simplify this expression...