here is a different approach to finding the 125th term

1,2,4,7,11,16,22,29,37,... is the sequence

the difference between the 8th and 9th terms is 8

the difference between the 7th and 8th term is 7

therefore the difference between the 124th term and the 125th term is 124

if we add the differences: 1+2=3, 1 less than the 3rd term

1+2+3=6, 1 less than the 4th term

1+2+3+4=10, 1 less than the 5th term

the difference between the 125th term and the 124th term is 124 by looking at the other differences

the sum of all the differences will be 1 less than the 125th term

so we add all the differences and add 1 to get the 125th term

so we add 1+2+3+4+5+6+...+119+120+121+122+123+124

add 1+124=125, 2+123=125, 3+122=125 and so on

how many groups of 125 do we have ? we have 62 groups

therefore 125*62=7750 and now we add 1 to 7750 to get 7751, the 125th term

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