Let the $n$ th term of the series $2 + 5 + 12 + 31 + 86 + . . .$ be $T_{n} = n + k^{n - 1}$ .Find $k$

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Solution

$2, 5, 12, 31, 81, . . . .$ (given series)
$3, 7, 19, 55, . . . . . .$ (first consecutive differences)
$4, 12, 36, . . . . .$ (second consecutive differences are in G.P with common ratio 3)
$∴ T_{n} = a + b n + c {.3}^{n - 1}$
$T_{1} = a + b + c = 2, T_{2} = a + 2 b + 3 c = 5$ and $T_{3} = a + 3 b + 9 c = 12$
then we get $a = 0, b = 1$ and $c = 1$
Hence, $T_{n} = n + 3^{n - 1}$

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Let the nth term of the series 2+5+12+31+86+... be Tn=n+kn−1.Find k

2,5,12,31,81,.... (given series) 3,7,19,55,...... (first consecutive differences) 4,12,36,..... (second consecutive differences are in G.P with common ratio 3)∴Tn=a+bn+c.3n−1T1=a+b+c=2,T2=a+2b+3c=5 and T3=a+3b+9c=12then we get a=0,b=1 and c=1Hence, Tn=n+3n−1

Let the $n$ th term of the series $2 + 5 + 12 + 31 + 86 + . . .$ be $T_{n} = n + k^{n - 1}$ .Find $k$