# Mental Math: Given the sequence -3,-2, 0, 3, 7, 12, 18,....,63, k, 88,... Find k=______

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I think the last two lines should say `-2k=-150,` and `k=75.`

A similar solution is to note that `k=63+n` for some `n,` and according to the pattern of increasing by one more than the previous increase,

`88=k+n+1=(63+n)+n+1=64+2n=>24=2n=>12=n,`

so `k=63+12=75.`

`T_1 = -3`

`T_2 = T_1+1 = -3+1 = -2`

`T_3 = T_2+2 = -2+2 = 0`

`T_4 = T_3+3 = 0+3 = 3`

Similarly we can say;

`T_(n+1) = T_(n)+n`

Let us say K is the (n+1) term. Then 63 will come as in nth term and 88 will come in (n+2).

`T_(n+1) = T_(n)+n`

`K =63+n ----(1)`

`T_(n+2) = T_(n+1)+(n+1)`

`88 = K+n+1 ----(2)`

`(2)-(1)`

`88-K = K+1-63`

`-2K = -150`

`K = 75`

*So the value of K is 75.*