Question

the sum of 40 terms of the series 1+2+3+4+5+8+7+16+9+...is​

Answer 1

The sum of 40 terms of the series 1 + 2 + 3 + 4 + 5 + 8 + 7 + 16 +9+..= $2^{21}$ + 398

Step-by-step explanation:

The given series has a combination of two series:

1 + 3 + 5 + 7 + . . . . . .  (20 terms)  are in AP and

2 + 4 + 8 + 16 + . . . . . . . . (20 terms) are in GP.

To find, the sum of 40 terms of the series 1 + 2 + 3 + 4 + 5 + 8 + 7 + 16 +9+.. = ?

1 + 3 + 5 + 7 + . . . . . .  (20 terms)  are in AP

Here, first term(a) = 1, common difference(d) = 3 - 1 = 2 and number of terms(n) = 20

$S_{n}=\dfrac{n}{2}(2a+(n-1)d)$

$S_{20}=\dfrac{20}{2}[2(1)+(20-1)2]$

$=10(2+38)$

= 400

2 + 4 + 8 + 16 + . . . . . . . . (20 terms) are in GP.

Here, first term(a) = 2, common ratio(r) = 2 and number of terms(n) = 20

$S_{n} =\dfrac{a(r^n-1)}{r-1}$

$S_{20} =\dfrac{2(2^{20}-1)}{2-1}$

$=2^{20+1}-2$

$=2^{21}-2$

The sum of 40 terms of the series 1 + 2 + 3 + 4 + 5 + 8 + 7 + 16 +9+..

= 400 + $2^{21}$ - 2

= $2^{21}$ + 398

Answer 2

Step-by-step explanation:

The sum of 40 terms of the series 1 + 2 + 3 + 4 + 5 + 8 + 7 + 16 +9+..= 2^{21}2

21

+ 398

Step-by-step explanation:

The given series has a combination of two series:

1 + 3 + 5 + 7 + . . . . . . (20 terms) are in AP and

2 + 4 + 8 + 16 + . . . . . . . . (20 terms) are in GP.

To find, the sum of 40 terms of the series 1 + 2 + 3 + 4 + 5 + 8 + 7 + 16 +9+.. = ?

1 + 3 + 5 + 7 + . . . . . . (20 terms) are in AP

Here, first term(a) = 1, common difference(d) = 3 - 1 = 2 and number of terms(n) = 20

S_{n}=\dfrac{n}{2}(2a+(n-1)d)S

n

=

2

n

(2a+(n−1)d)

S_{20}=\dfrac{20}{2}[2(1)+(20-1)2]S

20

=

2

20

[2(1)+(20−1)2]

=10(2+38)=10(2+38)

= 400

2 + 4 + 8 + 16 + . . . . . . . . (20 terms) are in GP.

Here, first term(a) = 2, common ratio(r) = 2 and number of terms(n) = 20

S_{n} =\dfrac{a(r^n-1)}{r-1}S

n

=

r−1

a(r

n

−1)

S_{20} =\dfrac{2(2^{20}-1)}{2-1}S

20

=

2−1

2(2

20

−1)

=2^{20+1}-2=2

20+1

−2

=2^{21}-2=2

21

−2

The sum of 40 terms of the series 1 + 2 + 3 + 4 + 5 + 8 + 7 + 16 +9+..

= 400 + 2^{21}2

21

- 2

= 2^{21}2

21

+ 398