# Insert into GP

Between numbers 5 and 640 insert as many numbers to form geometric progression so sum of the numbers you entered will be 630. How many numbers you must insert?

Result

m =  6

#### Solution:

$a_1 = 5 \ \\ a_n = 640 \ \\ s_n = 630+640+5 = 1275 \ \\ a_n= a_1 q^{n-1} \ \\ s_n= \dfrac{q^n-1}{q-1} \ \\ \ \\ 128 = q^{n-1} = q^n/q \ \\ 1275= \dfrac{q^n-1}{q-1} \ \\ \ \\ 1275= 5\cdot \dfrac{ 128 q - 1}{q-1} \ \\ 1275q - 1275 = 5\cdot (128 q - 1) \ \\ 635q = 1270 \ \\ q= 2 \ \\ \ \\ 128 = q^{n-1} \ \\ \ln 128 = (n-1) \ln 2 \ \\ n = 8 \ \\ \ \\ m = n-2 = 8-2 = 6$

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